Solve gx-fx=left(x-1right)left(x+1right)^2/x

$$gx-fx = \frac{ \left( x-1 \right) { \left(x+1 \right) }^{ 2 } }{ x }$$

Solve for f

$$f=g-x-1+\frac{1}{x}+\frac{1}{x^{2}}$$

$$x\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x.

$$gxx-fxx=\left(x-1\right)\left(x+1\right)^{2}$$

Multiply x and x to get x^{2}.

$$gx^{2}-fxx=\left(x-1\right)\left(x+1\right)^{2}$$

Multiply x and x to get x^{2}.

$$gx^{2}-fx^{2}=\left(x-1\right)\left(x+1\right)^{2}$$

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

$$gx^{2}-fx^{2}=\left(x-1\right)\left(x^{2}+2x+1\right)$$

Use the distributive property to multiply x-1 by x^{2}+2x+1 and combine like terms.

$$gx^{2}-fx^{2}=x^{3}+x^{2}-x-1$$

Subtract gx^{2} from both sides.

$$-fx^{2}=x^{3}+x^{2}-x-1-gx^{2}$$

Reorder the terms.

$$-fx^{2}=x^{3}-gx^{2}+x^{2}-x-1$$

The equation is in standard form.

$$\left(-x^{2}\right)f=x^{3}-gx^{2}+x^{2}-x-1$$

Divide both sides by -x^{2}.

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

Dividing by -x^{2} undoes the multiplication by -x^{2}.

$$f=\frac{x^{3}-gx^{2}+x^{2}-x-1}{-x^{2}}$$

Divide -x-1-gx^{2}+x^{2}+x^{3} by -x^{2}.

$$f=g-x-1+\frac{1}{x}+\frac{1}{x^{2}}$$

Solve for g

$$g=f+x+1-\frac{1}{x}-\frac{1}{x^{2}}$$

$$x\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x.

$$gxx-fxx=\left(x-1\right)\left(x+1\right)^{2}$$

Multiply x and x to get x^{2}.

$$gx^{2}-fxx=\left(x-1\right)\left(x+1\right)^{2}$$

Multiply x and x to get x^{2}.

$$gx^{2}-fx^{2}=\left(x-1\right)\left(x+1\right)^{2}$$

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

$$gx^{2}-fx^{2}=\left(x-1\right)\left(x^{2}+2x+1\right)$$

Use the distributive property to multiply x-1 by x^{2}+2x+1 and combine like terms.

$$gx^{2}-fx^{2}=x^{3}+x^{2}-x-1$$

Add fx^{2} to both sides.

$$gx^{2}=x^{3}+x^{2}-x-1+fx^{2}$$

The equation is in standard form.

$$x^{2}g=x^{3}+fx^{2}+x^{2}-x-1$$

Divide both sides by x^{2}.

$$\frac{x^{2}g}{x^{2}}=\frac{x^{3}+fx^{2}+x^{2}-x-1}{x^{2}}$$

Dividing by x^{2} undoes the multiplication by x^{2}.

$$g=\frac{x^{3}+fx^{2}+x^{2}-x-1}{x^{2}}$$

Divide x^{2}-x-1+fx^{2}+x^{3} by x^{2}.

$$g=f+x+1-\frac{x+1}{x^{2}}$$