Solve P(x)*P(x-1)=2x^2+3x+1

$$P ( x ) \cdot P ( x - 1 ) = 2 x ^ { 2 } + 3 x + 1$$

Solve for P (complex solution)

$$P=-ix^{-\frac{1}{2}}\left(x-1\right)^{-\frac{1}{2}}\sqrt{-x-1}\sqrt{2x+1}$$

$$P=ix^{-\frac{1}{2}}\left(x-1\right)^{-\frac{1}{2}}\sqrt{-x-1}\sqrt{2x+1}\text{, }x\neq 1\text{ and }x\neq 0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{P^{4}+10P^{2}+1}+P^{2}+3}{2\left(P^{2}-2\right)}\text{; }x=\frac{-\sqrt{P^{4}+10P^{2}+1}+P^{2}+3}{2\left(P^{2}-2\right)}\text{, }&P\neq -\sqrt{2}\text{ and }P\neq \sqrt{2}\\x=\frac{1}{-P^{2}-3}\text{, }&P=-\sqrt{2}\text{ or }P=\sqrt{2}\end{matrix}\right.$$

Solve for P

$$P=\sqrt{\frac{\left(x+1\right)\left(2x+1\right)}{x\left(x-1\right)}}$$

$$P=-\sqrt{\frac{\left(x+1\right)\left(2x+1\right)}{x\left(x-1\right)}}\text{, }x\leq -1\text{ or }\left(x\geq -\frac{1}{2}\text{ and }x<0\right)\text{ or }x>1$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{P^{4}+10P^{2}+1}+P^{2}+3}{2\left(P^{2}-2\right)}\text{; }x=\frac{-\sqrt{P^{4}+10P^{2}+1}+P^{2}+3}{2\left(P^{2}-2\right)}\text{, }&|P|\neq \sqrt{2}\\x=\frac{1}{-P^{2}-3}\text{, }&|P|=\sqrt{2}\end{matrix}\right.$$