Finite Math Examples

$\frac{3 x^{3} + 13 x - 4}{x^{3} + x^{2} - 2 x} = \frac{a}{x} + \frac{b}{x + 2} + \frac{c}{x - 1}$

Step 1

Rewrite the equation as $\frac{a}{x} + \frac{b}{x + 2} + \frac{c}{x - 1} = \frac{3 x^{3} + 13 x - 4}{x^{3} + x^{2} - 2 x}$ .

$\frac{a}{x} + \frac{b}{x + 2} + \frac{c}{x - 1} = \frac{3 x^{3} + 13 x - 4}{x^{3} + x^{2} - 2 x}$

Step 2

Move all terms containing variables to the left side of the equation.

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Step 2.1

Subtract $\frac{3 x^{3} + 13 x - 4}{x^{3} + x^{2} - 2 x}$ from both sides of the equation.

$\frac{a}{x} + \frac{b}{x + 2} + \frac{c}{x - 1} - \frac{3 x^{3} + 13 x - 4}{x^{3} + x^{2} - 2 x} = 0$

Step 2.2

Simplify the denominator.

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Step 2.2.1

Factor $x$ out of $x^{3} + x^{2} - 2 x$ .

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Step 2.2.1.1

Factor $x$ out of $x^{3}$ .

$\frac{a}{x} + \frac{b}{x + 2} + \frac{c}{x - 1} - \frac{3 x^{3} + 13 x - 4}{x \cdot x^{2} + x^{2} - 2 x} = 0$

Step 2.2.1.2

Factor $x$ out of $x^{2}$ .

$\frac{a}{x} + \frac{b}{x + 2} + \frac{c}{x - 1} - \frac{3 x^{3} + 13 x - 4}{x \cdot x^{2} + x \cdot x - 2 x} = 0$

Step 2.2.1.3

Factor $x$ out of $- 2 x$ .

$\frac{a}{x} + \frac{b}{x + 2} + \frac{c}{x - 1} - \frac{3 x^{3} + 13 x - 4}{x \cdot x^{2} + x \cdot x + x \cdot - 2} = 0$

Step 2.2.1.4

Factor $x$ out of $x \cdot x^{2} + x \cdot x$ .

$\frac{a}{x} + \frac{b}{x + 2} + \frac{c}{x - 1} - \frac{3 x^{3} + 13 x - 4}{x (x^{2} + x) + x \cdot - 2} = 0$

Step 2.2.1.5

Factor $x$ out of $x (x^{2} + x) + x \cdot - 2$ .

$\frac{a}{x} + \frac{b}{x + 2} + \frac{c}{x - 1} - \frac{3 x^{3} + 13 x - 4}{x (x^{2} + x - 2)} = 0$

Step 2.2.2

Factor $x^{2} + x - 2$ using the AC method.

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Step 2.2.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $1$ .

$- 1, 2$

Step 2.2.2.2

Write the factored form using these integers.

$\frac{a}{x} + \frac{b}{x + 2} + \frac{c}{x - 1} - \frac{3 x^{3} + 13 x - 4}{x ((x - 1) (x + 2))} = 0$

$\frac{a}{x} + \frac{b}{x + 2} + \frac{c}{x - 1} - \frac{3 x^{3} + 13 x - 4}{x (x - 1) (x + 2)} = 0$

Step 2.3

To write $\frac{a}{x}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

$\frac{a}{x} \cdot \frac{x + 2}{x + 2} + \frac{b}{x + 2} + \frac{c}{x - 1} - \frac{3 x^{3} + 13 x - 4}{x (x - 1) (x + 2)} = 0$

Step 2.4

To write $\frac{b}{x + 2}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{a}{x} \cdot \frac{x + 2}{x + 2} + \frac{b}{x + 2} \cdot \frac{x}{x} + \frac{c}{x - 1} - \frac{3 x^{3} + 13 x - 4}{x (x - 1) (x + 2)} = 0$

Step 2.5

Write each expression with a common denominator of $x (x + 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.5.1

Multiply $\frac{a}{x}$ by $\frac{x + 2}{x + 2}$ .

$\frac{a (x + 2)}{x (x + 2)} + \frac{b}{x + 2} \cdot \frac{x}{x} + \frac{c}{x - 1} - \frac{3 x^{3} + 13 x - 4}{x (x - 1) (x + 2)} = 0$

Step 2.5.2

Multiply $\frac{b}{x + 2}$ by $\frac{x}{x}$ .

$\frac{a (x + 2)}{x (x + 2)} + \frac{b x}{(x + 2) x} + \frac{c}{x - 1} - \frac{3 x^{3} + 13 x - 4}{x (x - 1) (x + 2)} = 0$

Step 2.5.3

Reorder the factors of $(x + 2) x$ .

$\frac{a (x + 2)}{x (x + 2)} + \frac{b x}{x (x + 2)} + \frac{c}{x - 1} - \frac{3 x^{3} + 13 x - 4}{x (x - 1) (x + 2)} = 0$

Step 2.6

Combine the numerators over the common denominator.

$\frac{a (x + 2) + b x}{x (x + 2)} + \frac{c}{x - 1} - \frac{3 x^{3} + 13 x - 4}{x (x - 1) (x + 2)} = 0$

Step 2.7

Simplify the numerator.

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Step 2.7.1

Apply the distributive property.

$\frac{a x + a \cdot 2 + b x}{x (x + 2)} + \frac{c}{x - 1} - \frac{3 x^{3} + 13 x - 4}{x (x - 1) (x + 2)} = 0$

Step 2.7.2

Move $2$ to the left of $a$ .

$\frac{a x + 2 a + b x}{x (x + 2)} + \frac{c}{x - 1} - \frac{3 x^{3} + 13 x - 4}{x (x - 1) (x + 2)} = 0$

Step 2.8

To write $\frac{a x + 2 a + b x}{x (x + 2)}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$\frac{a x + 2 a + b x}{x (x + 2)} \cdot \frac{x - 1}{x - 1} + \frac{c}{x - 1} - \frac{3 x^{3} + 13 x - 4}{x (x - 1) (x + 2)} = 0$

Step 2.9

To write $\frac{c}{x - 1}$ as a fraction with a common denominator, multiply by $\frac{x (x + 2)}{x (x + 2)}$ .

$\frac{a x + 2 a + b x}{x (x + 2)} \cdot \frac{x - 1}{x - 1} + \frac{c}{x - 1} \cdot \frac{x (x + 2)}{x (x + 2)} - \frac{3 x^{3} + 13 x - 4}{x (x - 1) (x + 2)} = 0$

Step 2.10

Write each expression with a common denominator of $x (x + 2) (x - 1)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.10.1

Multiply $\frac{a x + 2 a + b x}{x (x + 2)}$ by $\frac{x - 1}{x - 1}$ .

$\frac{(a x + 2 a + b x) (x - 1)}{x (x + 2) (x - 1)} + \frac{c}{x - 1} \cdot \frac{x (x + 2)}{x (x + 2)} - \frac{3 x^{3} + 13 x - 4}{x (x - 1) (x + 2)} = 0$

Step 2.10.2

Multiply $\frac{c}{x - 1}$ by $\frac{x (x + 2)}{x (x + 2)}$ .

$\frac{(a x + 2 a + b x) (x - 1)}{x (x + 2) (x - 1)} + \frac{c (x (x + 2))}{(x - 1) (x (x + 2))} - \frac{3 x^{3} + 13 x - 4}{x (x - 1) (x + 2)} = 0$

Step 2.10.3

Reorder the factors of $(x - 1) (x (x + 2))$ .

$\frac{(a x + 2 a + b x) (x - 1)}{x (x + 2) (x - 1)} + \frac{c (x (x + 2))}{x (x + 2) (x - 1)} - \frac{3 x^{3} + 13 x - 4}{x (x - 1) (x + 2)} = 0$

Step 2.11

Combine the numerators over the common denominator.

$\frac{(a x + 2 a + b x) (x - 1) + c (x (x + 2))}{x (x + 2) (x - 1)} - \frac{3 x^{3} + 13 x - 4}{x (x - 1) (x + 2)} = 0$

Step 2.12

Combine the numerators over the common denominator.

$\frac{(a x + 2 a + b x) (x - 1) + c (x (x + 2)) - (3 x^{3} + 13 x - 4)}{x (x + 2) (x - 1)} = 0$

Step 2.13

Simplify each term.

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Step 2.13.1

Expand $(a x + 2 a + b x) (x - 1)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{a x \cdot x + a x \cdot - 1 + 2 a x + 2 a \cdot - 1 + b x \cdot x + b x \cdot - 1 + c (x (x + 2)) - (3 x^{3} + 13 x - 4)}{x (x + 2) (x - 1)} = 0$

Step 2.13.2

Simplify each term.

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Step 2.13.2.1

Multiply $x$ by $x$ by adding the exponents.

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Step 2.13.2.1.1

Move $x$ .

$\frac{a (x \cdot x) + a x \cdot - 1 + 2 a x + 2 a \cdot - 1 + b x \cdot x + b x \cdot - 1 + c (x (x + 2)) - (3 x^{3} + 13 x - 4)}{x (x + 2) (x - 1)} = 0$

Step 2.13.2.1.2

Multiply $x$ by $x$ .

$\frac{a x^{2} + a x \cdot - 1 + 2 a x + 2 a \cdot - 1 + b x \cdot x + b x \cdot - 1 + c (x (x + 2)) - (3 x^{3} + 13 x - 4)}{x (x + 2) (x - 1)} = 0$

Step 2.13.2.2

Move $- 1$ to the left of $a x$ .

$\frac{a x^{2} - 1 \cdot (a x) + 2 a x + 2 a \cdot - 1 + b x \cdot x + b x \cdot - 1 + c (x (x + 2)) - (3 x^{3} + 13 x - 4)}{x (x + 2) (x - 1)} = 0$

Step 2.13.2.3

Rewrite $- 1 (a x)$ as $- (a x)$ .

$\frac{a x^{2} - (a x) + 2 a x + 2 a \cdot - 1 + b x \cdot x + b x \cdot - 1 + c (x (x + 2)) - (3 x^{3} + 13 x - 4)}{x (x + 2) (x - 1)} = 0$

Step 2.13.2.4

Multiply $- 1$ by $2$ .

$\frac{a x^{2} - a x + 2 a x - 2 a + b x \cdot x + b x \cdot - 1 + c (x (x + 2)) - (3 x^{3} + 13 x - 4)}{x (x + 2) (x - 1)} = 0$

Step 2.13.2.5

Multiply $x$ by $x$ by adding the exponents.

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Step 2.13.2.5.1

Move $x$ .

$\frac{a x^{2} - a x + 2 a x - 2 a + b (x \cdot x) + b x \cdot - 1 + c (x (x + 2)) - (3 x^{3} + 13 x - 4)}{x (x + 2) (x - 1)} = 0$

Step 2.13.2.5.2

Multiply $x$ by $x$ .

$\frac{a x^{2} - a x + 2 a x - 2 a + b x^{2} + b x \cdot - 1 + c (x (x + 2)) - (3 x^{3} + 13 x - 4)}{x (x + 2) (x - 1)} = 0$

Step 2.13.2.6

Move $- 1$ to the left of $b x$ .

$\frac{a x^{2} - a x + 2 a x - 2 a + b x^{2} - 1 \cdot (b x) + c (x (x + 2)) - (3 x^{3} + 13 x - 4)}{x (x + 2) (x - 1)} = 0$

Step 2.13.2.7

Rewrite $- 1 (b x)$ as $- (b x)$ .

$\frac{a x^{2} - a x + 2 a x - 2 a + b x^{2} - b x + c (x (x + 2)) - (3 x^{3} + 13 x - 4)}{x (x + 2) (x - 1)} = 0$

Step 2.13.3

Add $- a x$ and $2 a x$ .

$\frac{a x^{2} + 1 a x - 2 a + b x^{2} - b x + c (x (x + 2)) - (3 x^{3} + 13 x - 4)}{x (x + 2) (x - 1)} = 0$

Step 2.13.4

Multiply $a$ by $1$ .

$\frac{a x^{2} + a x - 2 a + b x^{2} - b x + c (x (x + 2)) - (3 x^{3} + 13 x - 4)}{x (x + 2) (x - 1)} = 0$

Step 2.13.5

Apply the distributive property.

$\frac{a x^{2} + a x - 2 a + b x^{2} - b x + c (x \cdot x + x \cdot 2) - (3 x^{3} + 13 x - 4)}{x (x + 2) (x - 1)} = 0$

Step 2.13.6

Multiply $x$ by $x$ .

$\frac{a x^{2} + a x - 2 a + b x^{2} - b x + c (x^{2} + x \cdot 2) - (3 x^{3} + 13 x - 4)}{x (x + 2) (x - 1)} = 0$

Step 2.13.7

Move $2$ to the left of $x$ .

$\frac{a x^{2} + a x - 2 a + b x^{2} - b x + c (x^{2} + 2 \cdot x) - (3 x^{3} + 13 x - 4)}{x (x + 2) (x - 1)} = 0$

Step 2.13.8

Apply the distributive property.

$\frac{a x^{2} + a x - 2 a + b x^{2} - b x + c x^{2} + c (2 x) - (3 x^{3} + 13 x - 4)}{x (x + 2) (x - 1)} = 0$

Step 2.13.9

Rewrite using the commutative property of multiplication.

$\frac{a x^{2} + a x - 2 a + b x^{2} - b x + c x^{2} + 2 c x - (3 x^{3} + 13 x - 4)}{x (x + 2) (x - 1)} = 0$

Step 2.13.10

Apply the distributive property.

$\frac{a x^{2} + a x - 2 a + b x^{2} - b x + c x^{2} + 2 c x - (3 x^{3}) - (13 x) - - 4}{x (x + 2) (x - 1)} = 0$

Step 2.13.11

Simplify.

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Step 2.13.11.1

Multiply $3$ by $- 1$ .

$\frac{a x^{2} + a x - 2 a + b x^{2} - b x + c x^{2} + 2 c x - 3 x^{3} - (13 x) - - 4}{x (x + 2) (x - 1)} = 0$

Step 2.13.11.2

Multiply $13$ by $- 1$ .

$\frac{a x^{2} + a x - 2 a + b x^{2} - b x + c x^{2} + 2 c x - 3 x^{3} - 13 x - - 4}{x (x + 2) (x - 1)} = 0$

Step 2.13.11.3

Multiply $- 1$ by $- 4$ .

$\frac{a x^{2} + a x - 2 a + b x^{2} - b x + c x^{2} + 2 c x - 3 x^{3} - 13 x + 4}{x (x + 2) (x - 1)} = 0$

Step 3

Set the numerator equal to zero.

$a x^{2} + a x - 2 a + b x^{2} - b x + c x^{2} + 2 c x - 3 x^{3} - 13 x + 4 = 0$

Step 4

Solve the equation for $a$ .

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Step 4.1

Move all terms not containing $a$ to the right side of the equation.

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Step 4.1.1

Subtract $b x^{2}$ from both sides of the equation.

$a x^{2} + a x - 2 a - b x + c x^{2} + 2 c x - 3 x^{3} - 13 x + 4 = - b x^{2}$

Step 4.1.2

Add $b x$ to both sides of the equation.

$a x^{2} + a x - 2 a + c x^{2} + 2 c x - 3 x^{3} - 13 x + 4 = - b x^{2} + b x$

Step 4.1.3

Subtract $c x^{2}$ from both sides of the equation.

$a x^{2} + a x - 2 a + 2 c x - 3 x^{3} - 13 x + 4 = - b x^{2} + b x - c x^{2}$

Step 4.1.4

Subtract $2 c x$ from both sides of the equation.

$a x^{2} + a x - 2 a - 3 x^{3} - 13 x + 4 = - b x^{2} + b x - c x^{2} - 2 c x$

Step 4.1.5

Add $3 x^{3}$ to both sides of the equation.

$a x^{2} + a x - 2 a - 13 x + 4 = - b x^{2} + b x - c x^{2} - 2 c x + 3 x^{3}$

Step 4.1.6

Add $13 x$ to both sides of the equation.

$a x^{2} + a x - 2 a + 4 = - b x^{2} + b x - c x^{2} - 2 c x + 3 x^{3} + 13 x$

Step 4.1.7

Subtract $4$ from both sides of the equation.

$a x^{2} + a x - 2 a = - b x^{2} + b x - c x^{2} - 2 c x + 3 x^{3} + 13 x - 4$

Step 4.2

Factor $a$ out of $a x^{2} + a x - 2 a$ .

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Step 4.2.1

Factor $a$ out of $a x^{2}$ .

$a (x^{2}) + a x - 2 a = - b x^{2} + b x - c x^{2} - 2 c x + 3 x^{3} + 13 x - 4$

Step 4.2.2

Factor $a$ out of $a x$ .

$a (x^{2}) + a (x) - 2 a = - b x^{2} + b x - c x^{2} - 2 c x + 3 x^{3} + 13 x - 4$

Step 4.2.3

Factor $a$ out of $- 2 a$ .

$a (x^{2}) + a (x) + a \cdot - 2 = - b x^{2} + b x - c x^{2} - 2 c x + 3 x^{3} + 13 x - 4$

Step 4.2.4

Factor $a$ out of $a (x^{2}) + a (x)$ .

$a (x^{2} + x) + a \cdot - 2 = - b x^{2} + b x - c x^{2} - 2 c x + 3 x^{3} + 13 x - 4$

Step 4.2.5

Factor $a$ out of $a (x^{2} + x) + a \cdot - 2$ .

$a (x^{2} + x - 2) = - b x^{2} + b x - c x^{2} - 2 c x + 3 x^{3} + 13 x - 4$

Step 4.3

Factor.

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Step 4.3.1

Factor $x^{2} + x - 2$ using the AC method.

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Step 4.3.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $1$ .

$- 1, 2$

Step 4.3.1.2

Write the factored form using these integers.

$a ((x - 1) (x + 2)) = - b x^{2} + b x - c x^{2} - 2 c x + 3 x^{3} + 13 x - 4$

Step 4.3.2

Remove unnecessary parentheses.

$a (x - 1) (x + 2) = - b x^{2} + b x - c x^{2} - 2 c x + 3 x^{3} + 13 x - 4$

Step 4.4

Divide each term in $a (x - 1) (x + 2) = - b x^{2} + b x - c x^{2} - 2 c x + 3 x^{3} + 13 x - 4$ by $(x - 1) (x + 2)$ and simplify.

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Step 4.4.1

Divide each term in $a (x - 1) (x + 2) = - b x^{2} + b x - c x^{2} - 2 c x + 3 x^{3} + 13 x - 4$ by $(x - 1) (x + 2)$ .

$\frac{a (x - 1) (x + 2)}{(x - 1) (x + 2)} = \frac{- b x^{2}}{(x - 1) (x + 2)} + \frac{b x}{(x - 1) (x + 2)} + \frac{- c x^{2}}{(x - 1) (x + 2)} + \frac{- 2 c x}{(x - 1) (x + 2)} + \frac{3 x^{3}}{(x - 1) (x + 2)} + \frac{13 x}{(x - 1) (x + 2)} + \frac{- 4}{(x - 1) (x + 2)}$

Step 4.4.2

Simplify the left side.

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Step 4.4.2.1

Cancel the common factor of $x - 1$ .

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Step 4.4.2.1.1

Cancel the common factor.

Step 4.4.2.1.2

Rewrite the expression.

$\frac{a (x + 2)}{x + 2} = \frac{- b x^{2}}{(x - 1) (x + 2)} + \frac{b x}{(x - 1) (x + 2)} + \frac{- c x^{2}}{(x - 1) (x + 2)} + \frac{- 2 c x}{(x - 1) (x + 2)} + \frac{3 x^{3}}{(x - 1) (x + 2)} + \frac{13 x}{(x - 1) (x + 2)} + \frac{- 4}{(x - 1) (x + 2)}$

Step 4.4.2.2

Cancel the common factor of $x + 2$ .

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Step 4.4.2.2.1

Cancel the common factor.

Step 4.4.2.2.2

Divide $a$ by $1$ .

$a = \frac{- b x^{2}}{(x - 1) (x + 2)} + \frac{b x}{(x - 1) (x + 2)} + \frac{- c x^{2}}{(x - 1) (x + 2)} + \frac{- 2 c x}{(x - 1) (x + 2)} + \frac{3 x^{3}}{(x - 1) (x + 2)} + \frac{13 x}{(x - 1) (x + 2)} + \frac{- 4}{(x - 1) (x + 2)}$

Step 4.4.3

Simplify the right side.

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Step 4.4.3.1

Simplify each term.

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Step 4.4.3.1.1

Move the negative in front of the fraction.

$a = - \frac{b x^{2}}{(x - 1) (x + 2)} + \frac{b x}{(x - 1) (x + 2)} + \frac{- c x^{2}}{(x - 1) (x + 2)} + \frac{- 2 c x}{(x - 1) (x + 2)} + \frac{3 x^{3}}{(x - 1) (x + 2)} + \frac{13 x}{(x - 1) (x + 2)} + \frac{- 4}{(x - 1) (x + 2)}$

Step 4.4.3.1.2

Move the negative in front of the fraction.

$a = - \frac{b x^{2}}{(x - 1) (x + 2)} + \frac{b x}{(x - 1) (x + 2)} - \frac{c x^{2}}{(x - 1) (x + 2)} + \frac{- 2 c x}{(x - 1) (x + 2)} + \frac{3 x^{3}}{(x - 1) (x + 2)} + \frac{13 x}{(x - 1) (x + 2)} + \frac{- 4}{(x - 1) (x + 2)}$

Step 4.4.3.1.3

Move the negative in front of the fraction.

$a = - \frac{b x^{2}}{(x - 1) (x + 2)} + \frac{b x}{(x - 1) (x + 2)} - \frac{c x^{2}}{(x - 1) (x + 2)} - \frac{2 c x}{(x - 1) (x + 2)} + \frac{3 x^{3}}{(x - 1) (x + 2)} + \frac{13 x}{(x - 1) (x + 2)} + \frac{- 4}{(x - 1) (x + 2)}$

Step 4.4.3.1.4

Move the negative in front of the fraction.

$a = - \frac{b x^{2}}{(x - 1) (x + 2)} + \frac{b x}{(x - 1) (x + 2)} - \frac{c x^{2}}{(x - 1) (x + 2)} - \frac{2 c x}{(x - 1) (x + 2)} + \frac{3 x^{3}}{(x - 1) (x + 2)} + \frac{13 x}{(x - 1) (x + 2)} - \frac{4}{(x - 1) (x + 2)}$