Solve f(x)=6x^3+6x^2+9xyg(x)=9x^3-3x^2+20

$$f ( x ) = 6 x ^ { 3 } + 6 x ^ { 2 } + 9 x y g ( x ) = 9 x ^ { 3 } - 3 x ^ { 2 } + 20$$

Solve for g

$$\left\{\begin{matrix}g=\frac{3x^{3}-9x^{2}+20}{9yx^{2}}\text{, }&x\neq 0\text{ and }y\neq 0\\g\in \mathrm{R}\text{, }&x=-\sqrt[3]{\frac{2\sqrt{10}+7}{3}}-\sqrt[3]{\frac{7-2\sqrt{10}}{3}}+1\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$6x^{3}+6x^{2}+9x^{2}yg=9x^{3}-3x^{2}+20$$

Subtract 6x^{3} from both sides.

$$6x^{2}+9x^{2}yg=9x^{3}-3x^{2}+20-6x^{3}$$

Combine 9x^{3} and -6x^{3} to get 3x^{3}.

$$6x^{2}+9x^{2}yg=3x^{3}-3x^{2}+20$$

Subtract 6x^{2} from both sides.

$$9x^{2}yg=3x^{3}-3x^{2}+20-6x^{2}$$

Combine -3x^{2} and -6x^{2} to get -9x^{2}.

$$9x^{2}yg=3x^{3}-9x^{2}+20$$

The equation is in standard form.

$$9yx^{2}g=3x^{3}-9x^{2}+20$$

Divide both sides by 9x^{2}y.

$$\frac{9yx^{2}g}{9yx^{2}}=\frac{3x^{3}-9x^{2}+20}{9yx^{2}}$$

Dividing by 9x^{2}y undoes the multiplication by 9x^{2}y.

$$g=\frac{3x^{3}-9x^{2}+20}{9yx^{2}}$$

Divide 3x^{3}-9x^{2}+20 by 9x^{2}y.

$$g=\frac{\frac{x^{3}}{3}-x^{2}+\frac{20}{9}}{yx^{2}}$$