CZ-SK-MO 2005 Problem 5

Let $p,q, r, s$ be real numbers with $q \ne -1$ and $s \ne -1$ . Prove that the quadratic equations $x^2 + px+q = 0$ and $x^2 +rx+s = 0$ have a common root, while their other roots are inverse of each other, if and only if $pr = (q+1)(s+1)$ and $p(q+1)s = r(s+1)q$ . (A double root is counted twice.)