CAPS 2004 Problem 1

Show that real numbers, $p, q, r$ satisfy the condition $p^4(q-r)^2 + 2p^2(q+r) + 1 = p^4$ if and only if the quadratic equations $x^2 + px + q = 0$ and $y^2 - py + r = 0$ have real roots (not necessarily distinct) which can be labeled by $x_1,x_2$ and $y_1,y_2$ , respectively, in such a way that $x_1y_1 - x_2y_2 = 1$ .