Solve f(x)=x^3-6x^2+11x-6g(x)=x+2

$$f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 11 x - 6 g ( x ) = x + 2$$

Solve for g

$$g=\frac{x^{2}}{6}-x+\frac{5}{3}-\frac{1}{3x}$$

$$x\neq 0$$

Steps for Solving Linear Equation

Subtract x^{3} from both sides.

$$-6x^{2}+11x-6gx=x+2-x^{3}$$

Add 6x^{2} to both sides.

$$11x-6gx=x+2-x^{3}+6x^{2}$$

Subtract 11x from both sides.

$$-6gx=x+2-x^{3}+6x^{2}-11x$$

Combine x and -11x to get -10x.

$$-6gx=-10x+2-x^{3}+6x^{2}$$

The equation is in standard form.

$$\left(-6x\right)g=2-10x+6x^{2}-x^{3}$$

Divide both sides by -6x.

$$\frac{\left(-6x\right)g}{-6x}=\frac{2-10x+6x^{2}-x^{3}}{-6x}$$

Dividing by -6x undoes the multiplication by -6x.

$$g=\frac{2-10x+6x^{2}-x^{3}}{-6x}$$

Divide -10x+2-x^{3}+6x^{2} by -6x.

$$g=\frac{x^{2}}{6}-x+\frac{5}{3}-\frac{1}{3x}$$