HMMT 2020 A9

Let $P(x)=x^{2020}+x+2$, which has 2020 distinct roots. Let $Q(x)$ be the monic polynomial of degree $\binom{2020}{2}$ whose roots are the pairwise products of the roots of $P(x)$. Let $\alpha$ satisfy $P(\alpha)=4$. Compute the sum of all possible values of $Q\left(\alpha^{2}\right)^{2}$.