Solve x^2+5x+7/(x+1)^2=A/(x+1)+B/(x+1)^2

$$\frac { x ^ { 2 } + 5 x + 7 } { ( x + 1 ) ^ { 2 } } = \frac { A } { ( x + 1 ) } + \frac { B } { ( x + 1 ) ^ { 2 } }$$

Solve for A

$$A=-\frac{-x^{2}-5x+B-7}{x+1}$$

$$x\neq -1$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x+1\right)^{2}, the least common multiple of \left(x+1\right)^{2},x+1.

$$x^{2}+5x+7=\left(x+1\right)A+B$$

Use the distributive property to multiply x+1 by A.

$$x^{2}+5x+7=xA+A+B$$

Swap sides so that all variable terms are on the left hand side.

$$xA+A+B=x^{2}+5x+7$$

Subtract B from both sides.

$$xA+A=x^{2}+5x+7-B$$

Combine all terms containing A.

$$\left(x+1\right)A=x^{2}+5x+7-B$$

The equation is in standard form.

$$\left(x+1\right)A=x^{2}+5x-B+7$$

Divide both sides by x+1.

$$\frac{\left(x+1\right)A}{x+1}=\frac{x^{2}+5x-B+7}{x+1}$$

Dividing by x+1 undoes the multiplication by x+1.

$$A=\frac{x^{2}+5x-B+7}{x+1}$$

Solve for B

$$B=x^{2}-Ax+5x-A+7$$

$$x\neq -1$$

Solution Steps

Multiply both sides of the equation by \left(x+1\right)^{2}, the least common multiple of \left(x+1\right)^{2},x+1.

$$x^{2}+5x+7=\left(x+1\right)A+B$$

Use the distributive property to multiply x+1 by A.

$$x^{2}+5x+7=xA+A+B$$

Swap sides so that all variable terms are on the left hand side.

$$xA+A+B=x^{2}+5x+7$$

Subtract xA from both sides.

$$A+B=x^{2}+5x+7-xA$$

Subtract A from both sides.

$$B=x^{2}+5x+7-xA-A$$