Solve 3x^2+2x+2/(x^2+1)^2=A/x

$$\frac { 3 x ^ { 2 } + 2 x + 2 } { ( x ^ { 2 } + 1 ) ^ { 2 } } = \frac { A } { x }$$

Solve for A

$$A=\frac{x\left(3x^{2}+2x+2\right)}{\left(x^{2}+1\right)^{2}}$$

$$x\neq 0$$

Steps for Solving Linear Equation

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

$$x\left(3x^{2}+2x+2\right)=\left(x^{2}+1\right)^{2}A$$

Use the distributive property to multiply x by 3x^{2}+2x+2.

$$3x^{3}+2x^{2}+2x=\left(x^{2}+1\right)^{2}A$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x^{2}+1\right)^{2}.

$$3x^{3}+2x^{2}+2x=\left(\left(x^{2}\right)^{2}+2x^{2}+1\right)A$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$3x^{3}+2x^{2}+2x=\left(x^{4}+2x^{2}+1\right)A$$

Use the distributive property to multiply x^{4}+2x^{2}+1 by A.

$$3x^{3}+2x^{2}+2x=x^{4}A+2x^{2}A+A$$

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

$$x^{4}A+2x^{2}A+A=3x^{3}+2x^{2}+2x$$

Combine all terms containing A.

$$\left(x^{4}+2x^{2}+1\right)A=3x^{3}+2x^{2}+2x$$

Divide both sides by x^{4}+2x^{2}+1.

$$\frac{\left(x^{4}+2x^{2}+1\right)A}{x^{4}+2x^{2}+1}=\frac{x\left(3x^{2}+2x+2\right)}{x^{4}+2x^{2}+1}$$

Dividing by x^{4}+2x^{2}+1 undoes the multiplication by x^{4}+2x^{2}+1.

$$A=\frac{x\left(3x^{2}+2x+2\right)}{x^{4}+2x^{2}+1}$$

Divide x\left(3x^{2}+2x+2\right) by x^{4}+2x^{2}+1.

$$A=\frac{x\left(3x^{2}+2x+2\right)}{\left(x^{2}+1\right)^{2}}$$