Find the value of x.
(6x + 1)°
(9x - 23)°
geometry

Answer:

Step-by-step explanation:

You want the steps to finding the equation of the hyperbola with center (1, 4), vertices (-7, 4) and (9, 4) and that includes the point (-11, -6).

The standard-form equation of a hyperbola with center (h, k) and semi-axes 'a' and 'b' is ...

  [tex]\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1[/tex]

The "linear eccentricity" 'c' is the distance from the center to a focus, and satisfies the equation ...

  c² = a² +b²

The vertices are (h±a, k) and the foci are (h±c, k).

The center of the hyperbola is given as (1, 4). The distance from the right vertex to the center is ...

  a = 9-1 = 8

The equation thus far is ...

  (x -1)/8² -(y -4)/b² = 1

The value of 'b' can be determined from the given point:

  (-11 -1)²/8² -(-6 -4)²/b² = 1

  9/4 -100/b² = 1

  5/4 = 100/b²

  b² = 100/(5/4) = 80

The linear eccentricity is ...

  c² = a² +b²

  c² = 64 +80 = 144

  c = √144 = 12

The foci are (1±12, 4) = (-11, 4) and (13, 4).

The equation is ...

  (x -1)²/64 -(y -4)²/80 = 1

Answer:

[tex]\dfrac{(x-1)^2}{64}-\dfrac{(y-4)^2}{80}=1[/tex]

Step-by-step explanation:

Standard equation of a horizontal hyperbola (opening left and right):

[tex]\boxed{\dfrac{(x-h)^2}{a^2}-\dfrac{(y-k)^2}{b^2}=1}[/tex]

where:

Given:

Therefore:

Find the value of a using the x-values of the vertices:

[tex]\begin{aligned}\implies h-a &= -7\\1-a &= -7\\a &= 8\end{aligned}[/tex]

[tex]\begin{aligned}\implies h+a &= 9\\1+a &= 9\\a &= 8\end{aligned}[/tex]

Therefore, a = 8.

Substitute the values of h, k and a into the formula:

[tex]\implies \dfrac{(x-1)^2}{8^2}-\dfrac{(y-4)^2}{b^2}=1[/tex]

[tex]\implies \dfrac{(x-1)^2}{64}-\dfrac{(y-4)^2}{b^2}=1[/tex]

Substitute the given point on the hyperbola (-11, -6) into the equation and solve for b²:

[tex]\implies \dfrac{(-11-1)^2}{64}-\dfrac{(-6-4)^2}{b^2}=1[/tex]

[tex]\implies \dfrac{(-12)^2}{64}-\dfrac{(-10)^2}{b^2}=1[/tex]

[tex]\implies \dfrac{144}{64}-\dfrac{100}{b^2}=1[/tex]

[tex]\implies \dfrac{144}{64}-1=\dfrac{100}{b^2}[/tex]

[tex]\implies \dfrac{144}{64}-\dfrac{64}{64}=\dfrac{100}{b^2}[/tex]

[tex]\implies \dfrac{80}{64}=\dfrac{100}{b^2}[/tex]

[tex]\implies \dfrac{5}{4}=\dfrac{100}{b^2}[/tex]

[tex]\implies b^2=\dfrac{4 \cdot 100}{5}[/tex]

[tex]\implies b^2=\dfrac{400}{5}[/tex]

[tex]\implies b^2=80[/tex]

Therefore, the equation of the hyperbola is:

[tex]\boxed{\dfrac{(x-1)^2}{64}-\dfrac{(y-4)^2}{80}=1}[/tex]

To find the foci, first find c where c² = a² + b²:

[tex]\implies c=\sqrt{a^2+b^2}[/tex]

[tex]\implies c=\sqrt{64+80}[/tex]

[tex]\implies c=\sqrt{144}[/tex]

[tex]\implies c=12[/tex]

Substitute the found value of c, along with the values of h and k into the foci formula:

[tex]\implies (h+c, k)=(1+12, 4)=(13,4)[/tex]

[tex]\implies (h-c,k)=(1-12,4)=(-11,4)[/tex]

Therefore, the foci are (-11, 4) and (13, 4).

The probability that the family had at least 3 girls is 0.6367

This is a binomial probability experiment. We are going to use the binomial distribution formula for determining the probability of x successes:

P(x = r) = nCr . p^r . q^n-r

Given: n= 8, probabibility of sucess (p) = 0.5, x≥3 (at least 3 girls)

The failures can be calculated using q = 1 - p = 1 - 0.5 = 0.5

P(x≥3) = 1 - P(x≤3) = 1 - ( P(x= 0) + P(x= 1) + P(x= 2) + P(x= 3) )

Using the formula:

P(x≥3)  = 1 - ( (8C0 x 0.5⁰  x 0.5⁸⁻⁰) +(8C1 x 0.5¹  x 0.5⁸⁻¹) + (8C2 x 0.5²  x 0.5⁸⁻²)  +  (8C3 x 0.5³  x 0.5⁸⁻³) )

          = 1 - ( (1/256) + (1/32) + (7/64) + (7/32) )

         = 0.6367

Therefore, the probability that the Wilson family had at least 3 girls is 0.6367

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The measure of central angle is 45°

Here, arc length S = 5π/4

radius r = 5

We need to find central angle θ

Using the formula of arc length,

S = r * θ

5π/4 = 5 * θ

θ = π/4

θ = 45°

Therefore, the measure of central angle is 45°

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Answer: 60 minutes

Step-by-step explanation:

x=time Madison spent at the mall

2x + 15 = 135

2x = 135-15

x = 120/2 = 60

The number which is as represented on the calculator display is; Choice A; 0.000001231151

It follows from the task content that the number which is represented on the calculator display is to be identified.

By checking, The display of the calculator is; 1.231151 E-⁶.

The number above can be represented as follows;

1.231151 × 10-⁶.

Since a negative exponent on the number ten indicates a shift leftwards, it follows that the number represented on the calculator display is;

0.000001231151.

Therefore, the correct number represented on the calculator display is; Choice A; 0.000001231151.

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Answer: A

Step-by-step explanation: A

A

The difference in the amount of money earned by Beth and Bartley is $40.25.

Given that, Beth makes $10.25/hour and Bartley makes $11.50 an hour.

The unitary method is a technique for solving a problem by first finding the value of a single unit, and then finding the necessary value by multiplying the single unit value.

If Beth worked for 23 hours

Money earned by Beth is

10.25×23=$235.75

If Bartley worked for 17 hours

Money earned by Bartley is

11.50×17 =$195.5

So, difference of money = 235.75-195.5

= $40.25

Hence, the difference in the amount of money earned by Beth and Bartley is $40.25.

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Using the proportionality theorem, the values are:

x = 3

y = 2

The proportional theorem states that if two triangles are similar to each other, then their corresponding side lengths have the same ratios.

The triangles above are similar to each other, therefore, their corresponding sides will have lengths that are proportional or have the same ratios.

Therefore:

3/1.5 = 6/x

Cross multiple

3x = (1.5)(6)

3x = 9

3x/3 = 9/3

x = 3

Find y:

3/1.5 = 4/y

3y = (1.5)(4)

3y = 6

3y/3 = 6/3

y = 2

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Hence its proven that f and g are inverse of one another

What is a Function?

A function in mathematics from a set X to a set Y allocates exactly one element of Y to each element of X. The sets X and Y are collectively referred to as the function's domain and codomain, respectively.

Given,

f(x)= 2x-3/2

g(x)= 2x+3/2

Step 1: show that fog=x

f(g(x))

[2((2x + 3) / 2) - 3]/2

[(2x + 3) - 3] / 2

(2x + 3 - 3) / 2

2x / 2

x , which is equals to x

Now, for Step 2: gof = x

g(f(x))

g((2x - 3)/2)

[2((2x - 3)/2) + 3]/2

[(2x - 3) + 3]/2

(2x - 3 + 3)/2

2x/2

x , which is equals to x

Since, both fog(x) and gof(x) equals to x

Hence, its proven that f and g are inverse of one another

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Answer: 1/8x - 5/6

Step-by-step explanation:

Let's move the similar terms of the problem together.

So first we have:

-3/4x + 7/8x

And then we have:

-1/3 - 1/2

So to add/subtract fractions, the denominators have to be equal. We can change the -3/4 into -6/8 by just multiplying by 2/2 which is just 1. (Anything multiplied by 1 is just itself). So then we have:

-6/8x + 7/8x ---> 1/8x

Next, the second part.

Since neither denominators (3 or 2) can turn into one another by multiplying them with a number, we'll just multiply them with the opposite denominator.
So then we have:

-1/3 * (2/2) = -2/6
-1/2 * (3/3) = -3/6

-2/6 - 3/6 = -5/6

So finally we have:

1/8x - 5/6

The distance travelled before their first gas stop is 76.68 miles.

A fraction consisting of a quotient and remainder is a mixed fraction. we can convert the mixed fraction to improper fraction by first dividing the numerator by denominator and then taking the quotient as whole number and remainder as the numerator of proper fraction keeping the denominator same.

Consider that the distance covered before their first stop for gas be 'x'

The Total distance travelled is 372 miles

WE have given that after stopping for gas they travelled  65 1/4 miles and then tripled their distance. The equation is;

x + 261/4 + 3x = 372

4x + 261/4 = 372

16x + 261 = 1488

16x = 1227

x = 76.68 miles

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Answer of the why did George Washington caution against the formation of political parties?

What potential risks do political parties pose towards democracy? Do you think that the country should have followed his advice are as follows :  

(a) He believes that the government was undermined by differences between the political parties. Additionally, he argues that "the alternate domination" of one party over another and concurrent attempts to exact revenge on their rivals have resulted in terrible atrocities and "is itself a frightful despotism."

(b)The democratic theory's main tenet is that democracies force administrations to take citizens' preferences into account. Every contemporary democracy is organized by political parties, and some observers contend that parties are what makes democracies responsive. Others, however, contend that parties give extremists a platform and make governments less receptive to the needs of the populace. As newly established democracies around the world grapple with concerns of representation and governability, the discussion of parties and democracy assumes fresh prominence.

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Answer:

the equation is 37.45-5.35v=0

Step-by-step explanation:

37.45-5.35v=0

minus 37.45 to both sides

-5.35v=-37.45

divide -5.35 from both sides

v=7

after 7 days she will finish the card

The equivalent multiplication sentence that shows associative property is 3x(1×6)

Associative property is defined as, when more than two numbers are added or multiplied, the result remains the same, irrespective of how they are grouped.

This means that if a,b,c are real numbers then a×(b×c)=b×(a×c)

using the associative property,1×(3×6)= 3×(1×6)

To verify the statement,let's find out

solving the parentheses first

1×(3×6)=1×18= 18

3×(1×6)= 3×6= 18

this means that 1×(3×6) and 3×(1×6) are associative

therefore 1×(3×6)=3×(1×6)

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Answer:

least

- 1/4 ft

- 14 3/4 ft

- 15 1/2 ft

- 20 3/5 ft

greatest

Answer:

- 20 3/5 ft

- 15 1/2 ft

- 14 3/4 ft

- 1/4 ft

I the number is larger looking but have a negative symbol, then it is smaller.

For example, -1 is greater than -6. Also -1/4 is greater than -20 3/5.

Trust me, I am a year ahead in school, so trust me.

I hope this helped.

Answer:god is good

Step-by-step explanation:if u put this 1 stars u hate god

if u give 5 stars u love god

The vertex of the equation y=x^2-12x+7 is (6,-29) and the graph is given below.

In the given question we have to find the graphing form of y=x^2-12x+7.

The given equation is y=x^2-12x+7.

To graph the given equation we firstly express that equation in the standard form of parabola.

y=a(x-h)^2+k

Add and subtract 36 in the given equation;

y=x^2-12x+36-36+7

y=x^2-2×6×x+(6)^2-29

y=(x-6)^2-29

The vertex of the given equation is (6,-29).

The graph of the given equation is given below:

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The probability that three cards have the same number, using the hypergeometric distribution, is of:

0.0448 = 4.48%.

The probability mass function for the hypergeometric distribution is defined as follows:

[tex]P(X = x) = h(x,N,n,k) = \frac{C_{k,x}C_{N-k,n-x}}{C_{N,n}}[/tex]

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

The parameters of the function are listed below:

In the context of this problem, the values of these parameters are given as follows:

N = 28, k = 4, n = 5, x = 3.

Hence for one color, the probability that three cards have the same number is:

[tex]P(X = x) = h(x,N,n,k) = \frac{C_{k,x}C_{N-k,n-x}}{C_{N,n}}[/tex]

[tex]P(X = 3) = h(3,28,5,4) = \frac{C_{4,3}C_{24,2}}{C_{28,5}} = 0.0112[/tex]

There are four colors, hence the probability is:

p = 4 x 0.0112 = 0.0448 = 4.48%.

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Answer:

slope: 1

y-intercept: (0,2)

the second point to plot: (1,3)

Step-by-step explanation:

Points to plot

(0,2),(1,3),(-2,0)

From the given pair of parallel lines , the lines which satisfies the condition of two angles to be congruent that is ∠5 ≅ ∠7 are l || m .

As given in the question,

Given pair of lines,

Condition given that two angles are congruent ,

∠5 ≅ ∠7

Angle 5 is congruent to vertical opposite angle and to make a pair of parallel line Opposite angle should be congruent to angle 7.

Opposite angle and Angle 7 are pair of corresponding angles.

Line l should be parallel to the line m to satisfied the condition of

∠5 ≅ ∠7.

Therefore, from the given pair of parallel lines , the lines which satisfies the condition of two angles to be congruent that is ∠5 ≅ ∠7 are l || m .

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The number of dinners that should be prepared to maximize profit is 1200.

The maximum profit is $1040

We know that,

y = ax²+ bx + c

The expression has the greatest value at x = -b/2a

From the question, we have

y=-0.001x² +2.4x-400

x = -b/2a

= -2.4/2*(-0.001)

=1200

substituting the value  we get

y=-0.001x² +2.4x-400

y=-0.001*1200² +2.4*1200-400

y=$1040

Multiplication:

Finding the product of two or more numbers in mathematics is done by multiplying the numbers. It is one of the fundamental operations in mathematics that we perform on a daily basis. Multiplication tables are the main use that is obvious. In mathematics, the repeated addition of one number in relation to another is represented by the multiplication of two numbers. These figures can be fractions, integers, whole numbers, natural numbers, etc. When m is multiplied by n, either m is added to itself 'n' times or the other way around.

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Answer: 48

Step-by-step explanation:  

         first you divide 25/2 and you get 12.5 and that is irrelevant but what is relevant is that you divide 96 by 2 and you get 48 thus giving you the answer to that question. :)

The domain is  {-5, -4, -2, 0, 1} and range is {4, -1, 1, 4, 3}

The given relation is {(−5,4), (−4, −1), (−2,1), (0,4), (1,3)

We need to find its domain and range

Domain of the set is {-5, -4, -2, 0, 1}

Range of the set is {4, -1, 1, 4, 3}

If in an relation every input has an single output, it is said to be a function

Here, every element of the domain has a single element in range, so it is a function

The given relation is  {(−3, −4), (−1,2), (0,0), (−3,5), (2,4)}

We need to find its domain and range

Domain of the set is {-3, -1, 0, -3, 2}

Range of the set is {-4, 2, 0, 5, 4}

If in an relation every input has an single output, it is said to be a function

Here, every element of the domain has a single element in range, so it is a function

Therefore, the domain is  {-5, -4, -2, 0, 1} and range is {4, -1, 1, 4, 3}

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Answer: 1. 0.1333 2. 1

Step-by-step explanation:

Step 1: 2/15 is the same as 2 divided by 15 which is 1.33333333333333333333333333333333333333333333333333 and so on.

Step 2: Clearly, the only number that is repeating is the 3.

Answer: This is a 1 by 5 rectangle

perimeter = 12 units

========================================================

Work Shown:

Label the vertices A,B,C,D

A =  (1, 4)

B = (1, 5)

C = (6, 5)

D = (6, 4)

The x coordinates of points A and B are the same (x = 1). The difference in y values of these points is 5-4 = 1 unit. This is the height of the rectangle. In other words, it's the length of segment AB.

Points B and C have the same y coordinate (y = 5). The difference in x value is 6-1 = 5 which is the horizontal length of the rectangle. This is the length of segment BC.

In short, this is a 1 by 5 rectangle. It's one unit tall and five units across.

--------

Perimeter = 2*(length+width)

Perimeter = 2*(5+1)

Perimeter = 2*6

Perimeter = 12 units

Or you can add up the four sides (1,5,1 and 5) to get the same result.

The value of the expression given as (7/4x - 5) - (2y - 3.5) - (-1/4x + 5) is 2x - 2y - 6.5

In this question, the representation of the expression is given as

(7/4x - 5) - (2y - 3.5) - (-1/4x + 5)

Start by removing the brackets in the expression

So, we have

7/4x - 5 - 2y + 3.5 + 1/4x - 5

Evaluate the fractions in the expression

The expression becomes

1.75x - 5 - 2y + 3.5 + 0.25x - 5

Collect the like terms

1.75x + 0.25x - 2y - 5  + 3.5  - 5

Evaluate the like terms

2x - 2y - 6.5

Hence, the solution is 2x - 2y - 6.5

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Answer:

  4.  121,800

  5.  591 3/32

  6.  1/4

Step-by-step explanation:

You want the sum of the first 200 terms of the arithmetic sequence starting 12, 18, 24, ....

The sum of the first n terms of an arithmetic sequence is given by the formula ...

  Sn = (2·a1 +d(n -1))(n/2)

where a1 is the first term and d is the common difference. Your sequence has first term 12 and common difference 18-12 = 6. The desired sum is ...

 S200 = (2·12 +6(200 -1))/(200/2) = (24 +1194)(100) = 121,800

You want the sum of the first 8 terms of the geometric sequence starting 12, 18, 27, ....

The sum of the first n terms of a geometric sequence is given by the formula ...

  Sn = a1·(r^n -1)/(r -1)

where a1 is the first term and r is the common ratio. Your sequence has first term 12 and common ratio 18/12 = 3/2. The desired sum is ...

  S8 = 12·((3/2)^8 -1)/(3/2 -1) = 24·6305/256 = 591 3/32

For this sequence, a1 = 1/12 and r = 2/3. When the sum is infinite and |r| < 1, the sum formula becomes ...

  S = a1/(1 -r)

The desired sum is ...

  S = (1/12)/(1 -2/3) = (1/12)/(4/12) = 1/4

Answer:

[tex]\textsf{4.} \quad 121,800[/tex]

[tex]\textsf{5.} \quad \dfrac{18915}{32}=591.09375[/tex]

[tex]\textsf{6.} \quad \dfrac{1}{4}[/tex]

Step-by-step explanation:

[tex]\boxed{\begin{minipage}{7.3 cm}\underline{Sum of the first $n$ terms of an arithmetic series}\\\\$S_n=\dfrac{1}{2}n[2a+(n-1)d]$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the first term. \\ \phantom{ww}$\bullet$ $d$ is the common difference.\\ \phantom{ww}$\bullet$ $n$ is the position of the term.\\\end{minipage}}[/tex]

Given arithmetic sequence:

The first term of the given sequence is 12.

The common difference can be found by subtracting the first term from the second term:

[tex]\implies d=a_2-a_1=18-12=6[/tex]

Therefore:

To find the sum of the first 200 terms, substitute n = 200, a = 12 and d = 6 into the formula:

[tex]\begin{aligned}S_{200}&=\dfrac{1}{2}(200)[2(12)+(200-1)(6)]\\&=100[24+(199)(6)]\\&=100[24+1194]\\&=100[1218]\\&=121800\end{aligned}[/tex]

[tex]\boxed{\begin{minipage}{7.3 cm}\underline{Sum of the first $n$ terms of a geometric series}\\\\$S_n=\dfrac{a(1-r^n)}{1-r}$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the first term. \\ \phantom{ww}$\bullet$ $r$ is the common ratio.\\ \phantom{ww}$\bullet$ $n$ is the position of the term.\\\end{minipage}}[/tex]

Given geometric sequence:

The first term of the given sequence is 12.

The common ratio can be found by dividing the second term by the first term:

[tex]\implies r=\dfrac{a_2}{a_1}=\dfrac{18}{12}=1.5[/tex]

Therefore:

To find the sum of the first 8 terms, substitute n = 8, a = 12 and r = 1.5 into the formula:

[tex]\begin{aligned}\implies S_{8}&=\dfrac{12(1-1.5^8)}{1-1.5}\\\\&=\dfrac{12\left(1-\frac{6561}{256}\right)}{-0.5}\\\\&=\dfrac{12\left(-\frac{6305}{256}\right)}{-0.5}\\\\&=\dfrac{-\frac{18915}{64}}{-0.5}\\\\&=\dfrac{18915}{32}\\\\&=591.09375\end{aligned}[/tex]

[tex]\boxed{\begin{minipage}{5.5 cm}\underline{Sum of an infinite geometric series}\\\\$S_{\infty}=\dfrac{a}{1-r}$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the first term. \\ \phantom{ww}$\bullet$ $r$ is the common ratio.\\\end{minipage}}[/tex]

Given geometric sequence:

The first term of the given sequence is ¹/₁₂.

The common ratio can be found by dividing the second term by the first term:

[tex]\implies r=\dfrac{a_2}{a_1}=\dfrac{\frac{1}{18}}{\frac{1}{12}}=\dfrac{2}{3}[/tex]

Therefore:

To find the sum of the infinite geometric sequence, substitute a = ¹/₁₂ and r = ²/₃ into the formula:

[tex]\begin{aligned}\implies S_{\infty}&=\dfrac{\frac{1}{12}}{1-\frac{2}{3}}\\\\&=\dfrac{\frac{1}{12}}{\frac{1}{3}}\\\\&=\dfrac{1}{12} \times \dfrac{3}{1}\\\\&=\dfrac{3}{12}\\\\&=\dfrac{1}{4}\end{aligned}[/tex]

The measure of the required angle ∠VUT = 80 degrees

Given : A triangle with angle measures as :

∠RTU = 50°

exterior angle ∠UVR = 120°

To find : ∠VUT

In the given triangle we can use the exterior angle sum property

which states that the angle opposite to the inside of the triangle and the angle outside the triangle

The angle outside the triangle is 120 degrees

120 = ∠VUT  + 50

thus on subtracting 50 both sides we will get

∠VUT = 120 - 50

∠VUT = 80 degrees

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The given rational expression when added gives

[tex] \frac{12x + 31}{(x + 3)(x - 4)(x + 2)} [/tex]

Given:

[tex] \frac{5}{(x + 3)(x - 4)} + \frac{7}{(x + 2)(x - 4)} [/tex]

=

[tex] \frac{5(x + 2) + 7(x + 3)}{(x + 3)(x - 4)(x + 2)} [/tex]

=

[tex] \frac{5x + 10 + 7x + 21}{(x + 3)(x - 4)(x + 2)} [/tex]

=

[tex] \frac{12x + 31}{(x + 3)(x - 4)(x + 2)} [/tex]

So therefore, the addition of the rational expression is

[tex] \frac{12x + 31}{(x + 3)(x - 4)(x + 2)} [/tex]

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