Solve f(x)+f(2)=x^3-m

$$f ( x ) + f ( 2 ) = x ^ { 3 } - m$$

Solve for f (complex solution)

$$\left\{\begin{matrix}f=-\frac{m-x^{3}}{x+2}\text{, }&x\neq -2\\f\in \mathrm{C}\text{, }&m=-8\text{ and }x=-2\end{matrix}\right.$$

Steps for Solving Linear Equation

Combine all terms containing f.

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

Divide both sides by x+2.

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

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

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

Solve for f

$$\left\{\begin{matrix}f=-\frac{m-x^{3}}{x+2}\text{, }&x\neq -2\\f\in \mathrm{R}\text{, }&x=-2\text{ and }m=-8\end{matrix}\right.$$

Steps for Solving Linear Equation

Combine all terms containing f.

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

Divide both sides by x+2.

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

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

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

Solve for m

$$m=x^{3}-fx-2f$$

Steps for Solving Linear Equation

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

$$x^{3}-m=fx+f\times 2$$

Subtract x^{3} from both sides.

$$-m=fx+f\times 2-x^{3}$$

The equation is in standard form.

$$-m=2f+fx-x^{3}$$

Divide both sides by -1.

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

Dividing by -1 undoes the multiplication by -1.

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

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

$$m=x^{3}-fx-2f$$