Solve A/x+2+B/x+3=x+4/x^2+5x+6

$$\frac { A } { x + 2 } + \frac { B } { x + 3 } = \frac { x + 4 } { x ^ { 2 } + 5 x + 6 }$$

Solve for A

$$A=-\frac{Bx-x+2B-4}{x+3}$$

$$x\neq -2\text{ and }x\neq -3$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x+2\right)\left(x+3\right), the least common multiple of x+2,x+3,x^{2}+5x+6.

$$\left(x+3\right)A+\left(x+2\right)B=x+4$$

Use the distributive property to multiply x+3 by A.

$$xA+3A+\left(x+2\right)B=x+4$$

Use the distributive property to multiply x+2 by B.

$$xA+3A+xB+2B=x+4$$

Subtract xB from both sides.

$$xA+3A+2B=x+4-xB$$

Subtract 2B from both sides.

$$xA+3A=x+4-xB-2B$$

Combine all terms containing A.

$$\left(x+3\right)A=x+4-xB-2B$$

The equation is in standard form.

$$\left(x+3\right)A=4-2B+x-Bx$$

Divide both sides by x+3.

$$\frac{\left(x+3\right)A}{x+3}=\frac{4-2B+x-Bx}{x+3}$$

Dividing by x+3 undoes the multiplication by x+3.

$$A=\frac{4-2B+x-Bx}{x+3}$$

Solve for B

$$B=-\frac{Ax-x+3A-4}{x+2}$$

$$x\neq -2\text{ and }x\neq -3$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x+2\right)\left(x+3\right), the least common multiple of x+2,x+3,x^{2}+5x+6.

$$\left(x+3\right)A+\left(x+2\right)B=x+4$$

Use the distributive property to multiply x+3 by A.

$$xA+3A+\left(x+2\right)B=x+4$$

Use the distributive property to multiply x+2 by B.

$$xA+3A+xB+2B=x+4$$

Subtract xA from both sides.

$$3A+xB+2B=x+4-xA$$

Subtract 3A from both sides.

$$xB+2B=x+4-xA-3A$$

Combine all terms containing B.

$$\left(x+2\right)B=x+4-xA-3A$$

The equation is in standard form.

$$\left(x+2\right)B=4-3A+x-Ax$$

Divide both sides by x+2.

$$\frac{\left(x+2\right)B}{x+2}=\frac{4-3A+x-Ax}{x+2}$$

Dividing by x+2 undoes the multiplication by x+2.

$$B=\frac{4-3A+x-Ax}{x+2}$$