SE-SKOLORNAS 1978 Problem 6

$p(x)$ is a polynomial of degree $n$ with leading coefficient $c$ , and $q(x)$ is a polynomial of degree $m$ with leading coefficient $c$ , such that $$p(x)^2 = \left(x^2 - 1\right)q(x)^2 + 1$$ Show that $p'(x) = nq(x)$ .