MathNet 2023-part-04 — 191 problems (EN–VI) Question 141

Suppose that $P(x)$ and $Q(x)$ are both quadratic polynomials with leading coefficient $1$ such that $P(P(x)-x) = (Q(x))^2$ for all real numbers $x$ and $P(2) = 0$. Find the sum of all possible values of $P(10)$.