What do you know about polynomials?

Calculus Level 4

There are $n$ polynomials $\left(P_1(x)..... P_n(x)\right)$ satisfying the following two equations:

$\begin{aligned} 8 \cdot P(x) &= (P'(x))^2\cdot P''(x) \\ P(0) &= 4. \end{aligned}$

Find the value of $\displaystyle \sum_{i=1}^n P_i(1).$