Solve P(x)^prime=3x^3+ax^2-2x+a+15

$$P ( x ) ^ { \prime } = 3 x ^ { 3 } + a x ^ { 2 } - 2 x + a + 15$$

Solve for a

$$a=\frac{-3x^{3}+2x+P-15}{x^{2}+1}$$

Steps for Solving Linear Equation

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

$$3x^{3}+ax^{2}-2x+a+15=P\frac{\mathrm{d}}{\mathrm{d}x}(x)$$

Subtract 3x^{3} from both sides.

$$ax^{2}-2x+a+15=P\frac{\mathrm{d}}{\mathrm{d}x}(x)-3x^{3}$$

Add 2x to both sides.

$$ax^{2}+a+15=P\frac{\mathrm{d}}{\mathrm{d}x}(x)-3x^{3}+2x$$

Subtract 15 from both sides.

$$ax^{2}+a=P\frac{\mathrm{d}}{\mathrm{d}x}(x)-3x^{3}+2x-15$$

Combine all terms containing a.

$$\left(x^{2}+1\right)a=P\frac{\mathrm{d}}{\mathrm{d}x}(x)-3x^{3}+2x-15$$

The equation is in standard form.

$$\left(x^{2}+1\right)a=-3x^{3}+2x+P-15$$

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

$$\frac{\left(x^{2}+1\right)a}{x^{2}+1}=\frac{-3x^{3}+2x+P-15}{x^{2}+1}$$

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

$$a=\frac{-3x^{3}+2x+P-15}{x^{2}+1}$$

Solve for P

$$P=3x^{3}+ax^{2}-2x+a+15$$