Solve (x+1)P(x)=x^3+4x^2-10x+a

$$( x + 1 ) P ( x ) = x ^ { 3 } + 4 x ^ { 2 } - 10 x + a$$

Solve for P (complex solution)

$$\left\{\begin{matrix}P=-\frac{-x^{3}-4x^{2}+10x-a}{x\left(x+1\right)}\text{, }&x\neq -1\text{ and }x\neq 0\\P\in \mathrm{C}\text{, }&\left(a=0\text{ and }x=0\right)\text{ or }\left(a=-13\text{ and }x=-1\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x+1 by P.

$$\left(xP+P\right)x=x^{3}+4x^{2}-10x+a$$

Use the distributive property to multiply xP+P by x.

$$Px^{2}+Px=x^{3}+4x^{2}-10x+a$$

Combine all terms containing P.

$$\left(x^{2}+x\right)P=x^{3}+4x^{2}-10x+a$$

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

$$\frac{\left(x^{2}+x\right)P}{x^{2}+x}=\frac{x^{3}+4x^{2}-10x+a}{x^{2}+x}$$

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

$$P=\frac{x^{3}+4x^{2}-10x+a}{x^{2}+x}$$

Divide a+x^{3}+4x^{2}-10x by x^{2}+x.

$$P=\frac{x^{3}+4x^{2}-10x+a}{x\left(x+1\right)}$$

Solve for P

$$\left\{\begin{matrix}P=-\frac{-x^{3}-4x^{2}+10x-a}{x\left(x+1\right)}\text{, }&x\neq -1\text{ and }x\neq 0\\P\in \mathrm{R}\text{, }&\left(a=0\text{ and }x=0\right)\text{ or }\left(a=-13\text{ and }x=-1\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x+1 by P.

$$\left(xP+P\right)x=x^{3}+4x^{2}-10x+a$$

Use the distributive property to multiply xP+P by x.

$$Px^{2}+Px=x^{3}+4x^{2}-10x+a$$

Combine all terms containing P.

$$\left(x^{2}+x\right)P=x^{3}+4x^{2}-10x+a$$

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

$$\frac{\left(x^{2}+x\right)P}{x^{2}+x}=\frac{x^{3}+4x^{2}-10x+a}{x^{2}+x}$$

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

$$P=\frac{x^{3}+4x^{2}-10x+a}{x^{2}+x}$$

Divide a+x^{3}+4x^{2}-10x by x^{2}+x.

$$P=\frac{x^{3}+4x^{2}-10x+a}{x\left(x+1\right)}$$

Solve for a

$$a=-x\left(x^{2}-Px+4x-P-10\right)$$