Precalculus Examples

$\frac{- 6 x^{2} - 11 x - 2}{(x + 2) {(x + 1)}^{2}}$

Step 1

Decompose the fraction and multiply through by the common denominator.

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Step 1.1

For each factor in the denominator, create a new fraction using the factor as the denominator, and an unknown value as the numerator. Since the factor in the denominator is linear, put a single variable in its place $A$ .

$\frac{A}{x + 2}$

Step 1.2

$\frac{A}{x + 2} + \frac{B}{{(x + 1)}^{2}}$

Step 1.3

$\frac{A}{x + 2} + \frac{B}{{(x + 1)}^{2}} + \frac{C}{x + 1}$

Step 1.4

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $(x + 2) {(x + 1)}^{2}$ .

$\frac{(- 6 x^{2} - 11 x - 2) (x + 2) {(x + 1)}^{2}}{(x + 2) {(x + 1)}^{2}} = \frac{(A) (x + 2) {(x + 1)}^{2}}{x + 2} + \frac{(B) (x + 2) {(x + 1)}^{2}}{{(x + 1)}^{2}} + \frac{(C) (x + 2) {(x + 1)}^{2}}{x + 1}$

Step 1.5

Cancel the common factor of $x + 2$ .

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Step 1.5.1

Cancel the common factor.

Step 1.5.2

Rewrite the expression.

$\frac{(- 6 x^{2} - 11 x - 2) {(x + 1)}^{2}}{{(x + 1)}^{2}} = \frac{(A) (x + 2) {(x + 1)}^{2}}{x + 2} + \frac{(B) (x + 2) {(x + 1)}^{2}}{{(x + 1)}^{2}} + \frac{(C) (x + 2) {(x + 1)}^{2}}{x + 1}$

Step 1.6

Cancel the common factor of ${(x + 1)}^{2}$ .

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Step 1.6.1

Cancel the common factor.

$\frac{(- 6 x^{2} - 11 x - 2) {(x + 1)}^{2}}{{(x + 1)}^{2}} = \frac{(A) (x + 2) {(x + 1)}^{2}}{x + 2} + \frac{(B) (x + 2) {(x + 1)}^{2}}{{(x + 1)}^{2}} + \frac{(C) (x + 2) {(x + 1)}^{2}}{x + 1}$

Step 1.6.2

Divide $- 6 x^{2} - 11 x - 2$ by $1$ .

$- 6 x^{2} - 11 x - 2 = \frac{(A) (x + 2) {(x + 1)}^{2}}{x + 2} + \frac{(B) (x + 2) {(x + 1)}^{2}}{{(x + 1)}^{2}} + \frac{(C) (x + 2) {(x + 1)}^{2}}{x + 1}$

Step 1.7

Simplify each term.

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Step 1.7.1

Cancel the common factor of $x + 2$ .

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Step 1.7.1.1

Cancel the common factor.

$- 6 x^{2} - 11 x - 2 = \frac{A (x + 2) {(x + 1)}^{2}}{x + 2} + \frac{(B) (x + 2) {(x + 1)}^{2}}{{(x + 1)}^{2}} + \frac{(C) (x + 2) {(x + 1)}^{2}}{x + 1}$

Step 1.7.1.2

Divide $(A) {(x + 1)}^{2}$ by $1$ .

$- 6 x^{2} - 11 x - 2 = (A) {(x + 1)}^{2} + \frac{(B) (x + 2) {(x + 1)}^{2}}{{(x + 1)}^{2}} + \frac{(C) (x + 2) {(x + 1)}^{2}}{x + 1}$

Step 1.7.2

Rewrite ${(x + 1)}^{2}$ as $(x + 1) (x + 1)$ .

$- 6 x^{2} - 11 x - 2 = A ((x + 1) (x + 1)) + \frac{(B) (x + 2) {(x + 1)}^{2}}{{(x + 1)}^{2}} + \frac{(C) (x + 2) {(x + 1)}^{2}}{x + 1}$

Step 1.7.3

Expand $(x + 1) (x + 1)$ using the FOIL Method.

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Step 1.7.3.1

Apply the distributive property.

$- 6 x^{2} - 11 x - 2 = A (x (x + 1) + 1 (x + 1)) + \frac{(B) (x + 2) {(x + 1)}^{2}}{{(x + 1)}^{2}} + \frac{(C) (x + 2) {(x + 1)}^{2}}{x + 1}$

Step 1.7.3.2

Apply the distributive property.

$- 6 x^{2} - 11 x - 2 = A (x \cdot x + x \cdot 1 + 1 (x + 1)) + \frac{(B) (x + 2) {(x + 1)}^{2}}{{(x + 1)}^{2}} + \frac{(C) (x + 2) {(x + 1)}^{2}}{x + 1}$

Step 1.7.3.3

Apply the distributive property.

$- 6 x^{2} - 11 x - 2 = A (x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1) + \frac{(B) (x + 2) {(x + 1)}^{2}}{{(x + 1)}^{2}} + \frac{(C) (x + 2) {(x + 1)}^{2}}{x + 1}$

Step 1.7.4

Simplify and combine like terms.

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Step 1.7.4.1

Simplify each term.

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Step 1.7.4.1.1

Multiply $x$ by $x$ .

$- 6 x^{2} - 11 x - 2 = A (x^{2} + x \cdot 1 + 1 x + 1 \cdot 1) + \frac{(B) (x + 2) {(x + 1)}^{2}}{{(x + 1)}^{2}} + \frac{(C) (x + 2) {(x + 1)}^{2}}{x + 1}$

Step 1.7.4.1.2

Multiply $x$ by $1$ .

$- 6 x^{2} - 11 x - 2 = A (x^{2} + x + 1 x + 1 \cdot 1) + \frac{(B) (x + 2) {(x + 1)}^{2}}{{(x + 1)}^{2}} + \frac{(C) (x + 2) {(x + 1)}^{2}}{x + 1}$

Step 1.7.4.1.3

Multiply $x$ by $1$ .

$- 6 x^{2} - 11 x - 2 = A (x^{2} + x + x + 1 \cdot 1) + \frac{(B) (x + 2) {(x + 1)}^{2}}{{(x + 1)}^{2}} + \frac{(C) (x + 2) {(x + 1)}^{2}}{x + 1}$

Step 1.7.4.1.4

Multiply $1$ by $1$ .

$- 6 x^{2} - 11 x - 2 = A (x^{2} + x + x + 1) + \frac{(B) (x + 2) {(x + 1)}^{2}}{{(x + 1)}^{2}} + \frac{(C) (x + 2) {(x + 1)}^{2}}{x + 1}$

Step 1.7.4.2

Add $x$ and $x$ .

$- 6 x^{2} - 11 x - 2 = A (x^{2} + 2 x + 1) + \frac{(B) (x + 2) {(x + 1)}^{2}}{{(x + 1)}^{2}} + \frac{(C) (x + 2) {(x + 1)}^{2}}{x + 1}$

Step 1.7.5

Apply the distributive property.

$- 6 x^{2} - 11 x - 2 = A x^{2} + A (2 x) + A \cdot 1 + \frac{(B) (x + 2) {(x + 1)}^{2}}{{(x + 1)}^{2}} + \frac{(C) (x + 2) {(x + 1)}^{2}}{x + 1}$

Step 1.7.6

Simplify.

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Step 1.7.6.1

Rewrite using the commutative property of multiplication.

$- 6 x^{2} - 11 x - 2 = A x^{2} + 2 A x + A \cdot 1 + \frac{(B) (x + 2) {(x + 1)}^{2}}{{(x + 1)}^{2}} + \frac{(C) (x + 2) {(x + 1)}^{2}}{x + 1}$

Step 1.7.6.2

Multiply $A$ by $1$ .

$- 6 x^{2} - 11 x - 2 = A x^{2} + 2 A x + A + \frac{(B) (x + 2) {(x + 1)}^{2}}{{(x + 1)}^{2}} + \frac{(C) (x + 2) {(x + 1)}^{2}}{x + 1}$

Step 1.7.7

Cancel the common factor of ${(x + 1)}^{2}$ .

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Step 1.7.7.1

Cancel the common factor.

$- 6 x^{2} - 11 x - 2 = A x^{2} + 2 A x + A + \frac{(B) (x + 2) {(x + 1)}^{2}}{{(x + 1)}^{2}} + \frac{(C) (x + 2) {(x + 1)}^{2}}{x + 1}$

Step 1.7.7.2

Divide $(B) (x + 2)$ by $1$ .

$- 6 x^{2} - 11 x - 2 = A x^{2} + 2 A x + A + (B) (x + 2) + \frac{(C) (x + 2) {(x + 1)}^{2}}{x + 1}$

Step 1.7.8

Apply the distributive property.

$- 6 x^{2} - 11 x - 2 = A x^{2} + 2 A x + A + B x + B \cdot 2 + \frac{(C) (x + 2) {(x + 1)}^{2}}{x + 1}$

Step 1.7.9

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

$- 6 x^{2} - 11 x - 2 = A x^{2} + 2 A x + A + B x + 2 \cdot B + \frac{(C) (x + 2) {(x + 1)}^{2}}{x + 1}$

Step 1.7.10

Cancel the common factor of ${(x + 1)}^{2}$ and $x + 1$ .

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Step 1.7.10.1

Factor $x + 1$ out of $(C) (x + 2) {(x + 1)}^{2}$ .

$- 6 x^{2} - 11 x - 2 = A x^{2} + 2 A x + A + B x + 2 B + \frac{(x + 1) ((C) (x + 2) (x + 1))}{x + 1}$

Step 1.7.10.2

Cancel the common factors.

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Step 1.7.10.2.1

Multiply by $1$ .

$- 6 x^{2} - 11 x - 2 = A x^{2} + 2 A x + A + B x + 2 B + \frac{(x + 1) ((C) (x + 2) (x + 1))}{(x + 1) \cdot 1}$

Step 1.7.10.2.2

Cancel the common factor.

$- 6 x^{2} - 11 x - 2 = A x^{2} + 2 A x + A + B x + 2 B + \frac{(x + 1) ((C) (x + 2) (x + 1))}{(x + 1) \cdot 1}$

Step 1.7.10.2.3

Rewrite the expression.

$- 6 x^{2} - 11 x - 2 = A x^{2} + 2 A x + A + B x + 2 B + \frac{(C) (x + 2) (x + 1)}{1}$

Step 1.7.10.2.4

Divide $(C) (x + 2) (x + 1)$ by $1$ .

$- 6 x^{2} - 11 x - 2 = A x^{2} + 2 A x + A + B x + 2 B + (C) (x + 2) (x + 1)$

Step 1.7.11

Apply the distributive property.

$- 6 x^{2} - 11 x - 2 = A x^{2} + 2 A x + A + B x + 2 B + (C x + C \cdot 2) (x + 1)$

Step 1.7.12

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

$- 6 x^{2} - 11 x - 2 = A x^{2} + 2 A x + A + B x + 2 B + (C x + 2 \cdot C) (x + 1)$

Step 1.7.13

Expand $(C x + 2 C) (x + 1)$ using the FOIL Method.

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Step 1.7.13.1

Apply the distributive property.

$- 6 x^{2} - 11 x - 2 = A x^{2} + 2 A x + A + B x + 2 B + C x (x + 1) + 2 C (x + 1)$

Step 1.7.13.2

Apply the distributive property.

$- 6 x^{2} - 11 x - 2 = A x^{2} + 2 A x + A + B x + 2 B + C x \cdot x + C x \cdot 1 + 2 C (x + 1)$

Step 1.7.13.3

Apply the distributive property.

$- 6 x^{2} - 11 x - 2 = A x^{2} + 2 A x + A + B x + 2 B + C x \cdot x + C x \cdot 1 + 2 C x + 2 C \cdot 1$

Step 1.7.14

Simplify and combine like terms.

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Step 1.7.14.1

Simplify each term.

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Step 1.7.14.1.1

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

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Step 1.7.14.1.1.1

Move $x$ .

$- 6 x^{2} - 11 x - 2 = A x^{2} + 2 A x + A + B x + 2 B + C (x \cdot x) + C x \cdot 1 + 2 C x + 2 C \cdot 1$

Step 1.7.14.1.1.2

Multiply $x$ by $x$ .

$- 6 x^{2} - 11 x - 2 = A x^{2} + 2 A x + A + B x + 2 B + C x^{2} + C x \cdot 1 + 2 C x + 2 C \cdot 1$

Step 1.7.14.1.2

Multiply $C$ by $1$ .

$- 6 x^{2} - 11 x - 2 = A x^{2} + 2 A x + A + B x + 2 B + C x^{2} + C x + 2 C x + 2 C \cdot 1$

Step 1.7.14.1.3

Multiply $2$ by $1$ .

$- 6 x^{2} - 11 x - 2 = A x^{2} + 2 A x + A + B x + 2 B + C x^{2} + C x + 2 C x + 2 C$

Step 1.7.14.2

Add $C x$ and $2 C x$ .

$- 6 x^{2} - 11 x - 2 = A x^{2} + 2 A x + A + B x + 2 B + C x^{2} + 3 C x + 2 C$

Step 1.8

Simplify the expression.

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Step 1.8.1

Move $A$ .

$- 6 x^{2} - 11 x - 2 = A x^{2} + 2 x A + A + B x + 2 B + C x^{2} + 3 C x + 2 C$

Step 1.8.2

Reorder $B$ and $x$ .

$- 6 x^{2} - 11 x - 2 = A x^{2} + 2 x A + A + B x + 2 B + C x^{2} + 3 C x + 2 C$

Step 1.8.3

Move $2 B$ .

$- 6 x^{2} - 11 x - 2 = A x^{2} + 2 x A + A + B x + C x^{2} + 3 C x + 2 B + 2 C$

Step 1.8.4

Move $A$ .

$- 6 x^{2} - 11 x - 2 = A x^{2} + 2 x A + B x + C x^{2} + 3 C x + A + 2 B + 2 C$

Step 1.8.5

Move $B x$ .

$- 6 x^{2} - 11 x - 2 = A x^{2} + 2 x A + C x^{2} + B x + 3 C x + A + 2 B + 2 C$

Step 1.8.6

Move $2 x A$ .

$- 6 x^{2} - 11 x - 2 = A x^{2} + C x^{2} + 2 x A + B x + 3 C x + A + 2 B + 2 C$

Step 2

Create equations for the partial fraction variables and use them to set up a system of equations.

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Step 2.1

Create an equation for the partial fraction variables by equating the coefficients of $x^{2}$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$- 6 = A + C$

Step 2.2

Create an equation for the partial fraction variables by equating the coefficients of $x$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$- 11 = 2 A + B + 3 C$

Step 2.3

Create an equation for the partial fraction variables by equating the coefficients of the terms not containing $x$ . For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$- 2 = A + 2 B + 2 C$

Step 2.4

Set up the system of equations to find the coefficients of the partial fractions.

$- 6 = A + C$

$- 11 = 2 A + B + 3 C$

$- 2 = A + 2 B + 2 C$

$- 6 = A + C$

$- 11 = 2 A + B + 3 C$

$- 2 = A + 2 B + 2 C$

Step 3

Solve the system of equations.

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Step 3.1

Solve for $A$ in $- 6 = A + C$ .

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Step 3.1.1

Rewrite the equation as $A + C = - 6$ .

$A + C = - 6$

$- 11 = 2 A + B + 3 C$

$- 2 = A + 2 B + 2 C$

Step 3.1.2

Subtract $C$ from both sides of the equation.

$A = - 6 - C$

$- 11 = 2 A + B + 3 C$

$- 2 = A + 2 B + 2 C$

$A = - 6 - C$

$- 11 = 2 A + B + 3 C$

$- 2 = A + 2 B + 2 C$

Step 3.2

Replace all occurrences of $A$ with $- 6 - C$ in each equation.

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Step 3.2.1

Replace all occurrences of $A$ in $- 11 = 2 A + B + 3 C$ with $- 6 - C$ .

$- 11 = 2 (- 6 - C) + B + 3 C$

$A = - 6 - C$

$- 2 = A + 2 B + 2 C$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $2 (- 6 - C) + B + 3 C$ .

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Step 3.2.2.1.1

Simplify each term.

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Step 3.2.2.1.1.1

Apply the distributive property.

$- 11 = 2 \cdot - 6 + 2 (- C) + B + 3 C$

$A = - 6 - C$

$- 2 = A + 2 B + 2 C$

Step 3.2.2.1.1.2

Multiply $2$ by $- 6$ .

$- 11 = - 12 + 2 (- C) + B + 3 C$

$A = - 6 - C$

$- 2 = A + 2 B + 2 C$

Step 3.2.2.1.1.3

Multiply $- 1$ by $2$ .

$- 11 = - 12 - 2 C + B + 3 C$

$A = - 6 - C$

$- 2 = A + 2 B + 2 C$

$- 11 = - 12 - 2 C + B + 3 C$

$A = - 6 - C$

$- 2 = A + 2 B + 2 C$

Step 3.2.2.1.2

Add $- 2 C$ and $3 C$ .

$- 11 = - 12 + B + C$

$A = - 6 - C$

$- 2 = A + 2 B + 2 C$

$- 11 = - 12 + B + C$

$A = - 6 - C$

$- 2 = A + 2 B + 2 C$

$- 11 = - 12 + B + C$

$A = - 6 - C$

$- 2 = A + 2 B + 2 C$

Step 3.2.3

Replace all occurrences of $A$ in $- 2 = A + 2 B + 2 C$ with $- 6 - C$ .

$- 2 = (- 6 - C) + 2 B + 2 C$

$- 11 = - 12 + B + C$

$A = - 6 - C$

Step 3.2.4

Simplify the right side.

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Step 3.2.4.1

Add $- C$ and $2 C$ .

$- 2 = - 6 + 2 B + C$

$- 11 = - 12 + B + C$

$A = - 6 - C$

$- 2 = - 6 + 2 B + C$

$- 11 = - 12 + B + C$

$A = - 6 - C$

$- 2 = - 6 + 2 B + C$

$- 11 = - 12 + B + C$

$A = - 6 - C$

Step 3.3

Reorder $- 6$ and $- C$ .

$A = - C - 6$

$- 2 = - 6 + 2 B + C$

$- 11 = - 12 + B + C$

Step 3.4

Solve for $C$ in $- 2 = - 6 + 2 B + C$ .

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Step 3.4.1

Rewrite the equation as $- 6 + 2 B + C = - 2$ .

$- 6 + 2 B + C = - 2$

$A = - C - 6$

$- 11 = - 12 + B + C$

Step 3.4.2

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

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Step 3.4.2.1

Add $6$ to both sides of the equation.

$2 B + C = - 2 + 6$

$A = - C - 6$

$- 11 = - 12 + B + C$

Step 3.4.2.2

Subtract $2 B$ from both sides of the equation.

$C = - 2 + 6 - 2 B$

$A = - C - 6$

$- 11 = - 12 + B + C$

Step 3.4.2.3

Add $- 2$ and $6$ .

$C = 4 - 2 B$

$A = - C - 6$

$- 11 = - 12 + B + C$

$C = 4 - 2 B$

$A = - C - 6$

$- 11 = - 12 + B + C$

$C = 4 - 2 B$

$A = - C - 6$

$- 11 = - 12 + B + C$

Step 3.5

Replace all occurrences of $C$ with $4 - 2 B$ in each equation.

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Step 3.5.1

Replace all occurrences of $C$ in $A = - C - 6$ with $4 - 2 B$ .

$A = - (4 - 2 B) - 6$

$C = 4 - 2 B$

$- 11 = - 12 + B + C$

Step 3.5.2

Simplify the right side.

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Step 3.5.2.1

Simplify $- (4 - 2 B) - 6$ .

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Step 3.5.2.1.1

Simplify each term.

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Step 3.5.2.1.1.1

Apply the distributive property.

$A = - 1 \cdot 4 - (- 2 B) - 6$

$C = 4 - 2 B$

$- 11 = - 12 + B + C$

Step 3.5.2.1.1.2

Multiply $- 1$ by $4$ .

$A = - 4 - (- 2 B) - 6$

$C = 4 - 2 B$

$- 11 = - 12 + B + C$

Step 3.5.2.1.1.3

Multiply $- 2$ by $- 1$ .

$A = - 4 + 2 B - 6$

$C = 4 - 2 B$

$- 11 = - 12 + B + C$

$A = - 4 + 2 B - 6$

$C = 4 - 2 B$

$- 11 = - 12 + B + C$

Step 3.5.2.1.2

Subtract $6$ from $- 4$ .

$A = 2 B - 10$

$C = 4 - 2 B$

$- 11 = - 12 + B + C$

$A = 2 B - 10$

$C = 4 - 2 B$

$- 11 = - 12 + B + C$

$A = 2 B - 10$

$C = 4 - 2 B$

$- 11 = - 12 + B + C$

Step 3.5.3

Replace all occurrences of $C$ in $- 11 = - 12 + B + C$ with $4 - 2 B$ .

$- 11 = - 12 + B + 4 - 2 B$

$A = 2 B - 10$

$C = 4 - 2 B$

Step 3.5.4

Simplify $- 11 = - 12 + B + 4 - 2 B$ .

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Step 3.5.4.1

Simplify the left side.

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Step 3.5.4.1.1

Remove parentheses.

$- 11 = - 12 + B + 4 - 2 B$

$A = 2 B - 10$

$C = 4 - 2 B$

$- 11 = - 12 + B + 4 - 2 B$

$A = 2 B - 10$

$C = 4 - 2 B$

Step 3.5.4.2

Simplify the right side.

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Step 3.5.4.2.1

Simplify $- 12 + B + 4 - 2 B$ .

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Step 3.5.4.2.1.1

Add $- 12$ and $4$ .

$- 11 = B - 8 - 2 B$

$A = 2 B - 10$

$C = 4 - 2 B$

Step 3.5.4.2.1.2

Subtract $2 B$ from $B$ .

$- 11 = - B - 8$

$A = 2 B - 10$

$C = 4 - 2 B$

$- 11 = - B - 8$

$A = 2 B - 10$

$C = 4 - 2 B$

$- 11 = - B - 8$

$A = 2 B - 10$

$C = 4 - 2 B$

$- 11 = - B - 8$

$A = 2 B - 10$

$C = 4 - 2 B$

$- 11 = - B - 8$

$A = 2 B - 10$

$C = 4 - 2 B$

Step 3.6

Solve for $B$ in $- 11 = - B - 8$ .

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Step 3.6.1

Rewrite the equation as $- B - 8 = - 11$ .

$- B - 8 = - 11$

$A = 2 B - 10$

$C = 4 - 2 B$

Step 3.6.2

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

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Step 3.6.2.1

Add $8$ to both sides of the equation.

$- B = - 11 + 8$

$A = 2 B - 10$

$C = 4 - 2 B$

Step 3.6.2.2

Add $- 11$ and $8$ .

$- B = - 3$

$A = 2 B - 10$

$C = 4 - 2 B$

$- B = - 3$

$A = 2 B - 10$

$C = 4 - 2 B$

Step 3.6.3

Divide each term in $- B = - 3$ by $- 1$ and simplify.

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Step 3.6.3.1

Divide each term in $- B = - 3$ by $- 1$ .

$\frac{- B}{- 1} = \frac{- 3}{- 1}$

$A = 2 B - 10$

$C = 4 - 2 B$

Step 3.6.3.2

Simplify the left side.

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Step 3.6.3.2.1

Dividing two negative values results in a positive value.

$\frac{B}{1} = \frac{- 3}{- 1}$

$A = 2 B - 10$

$C = 4 - 2 B$

Step 3.6.3.2.2

Divide $B$ by $1$ .

$B = \frac{- 3}{- 1}$

$A = 2 B - 10$

$C = 4 - 2 B$

$B = \frac{- 3}{- 1}$

$A = 2 B - 10$

$C = 4 - 2 B$

Step 3.6.3.3

Simplify the right side.

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Step 3.6.3.3.1

Divide $- 3$ by $- 1$ .

$B = 3$

$A = 2 B - 10$

$C = 4 - 2 B$

$B = 3$

$A = 2 B - 10$

$C = 4 - 2 B$

$B = 3$

$A = 2 B - 10$

$C = 4 - 2 B$

$B = 3$

$A = 2 B - 10$

$C = 4 - 2 B$

Step 3.7

Replace all occurrences of $B$ with $3$ in each equation.

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Step 3.7.1

Replace all occurrences of $B$ in $A = 2 B - 10$ with $3$ .

$A = 2 (3) - 10$

$B = 3$

$C = 4 - 2 B$

Step 3.7.2

Simplify the right side.

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Step 3.7.2.1

Simplify $2 (3) - 10$ .

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Step 3.7.2.1.1

Multiply $2$ by $3$ .

$A = 6 - 10$

$B = 3$

$C = 4 - 2 B$

Step 3.7.2.1.2

Subtract $10$ from $6$ .

$A = - 4$

$B = 3$

$C = 4 - 2 B$

$A = - 4$

$B = 3$

$C = 4 - 2 B$

$A = - 4$

$B = 3$

$C = 4 - 2 B$

Step 3.7.3

Replace all occurrences of $B$ in $C = 4 - 2 B$ with $3$ .

$C = 4 - 2 \cdot 3$

$A = - 4$

$B = 3$

Step 3.7.4

Simplify the right side.

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Step 3.7.4.1

Simplify $4 - 2 \cdot 3$ .

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Step 3.7.4.1.1

Multiply $- 2$ by $3$ .

$C = 4 - 6$

$A = - 4$

$B = 3$

Step 3.7.4.1.2

Subtract $6$ from $4$ .

$C = - 2$

$A = - 4$

$B = 3$

$C = - 2$

$A = - 4$

$B = 3$

$C = - 2$

$A = - 4$

$B = 3$

$C = - 2$

$A = - 4$

$B = 3$

Step 3.8

List all of the solutions.

$C = - 2, A = - 4, B = 3$

Step 4

Replace each of the partial fraction coefficients in $\frac{A}{x + 2} + \frac{B}{{(x + 1)}^{2}} + \frac{C}{x + 1}$ with the values found for $A$ , $B$ , and $C$ .

$\frac{- 4}{x + 2} + \frac{3}{{(x + 1)}^{2}} + \frac{- 2}{x + 1}$