VMO 2003 Problem 5

Define p(x) = $4x^3$ - $2x^2$ - 15x + 9, q(x) = $12x^3$ + $6x^2$ - 7x + 1. Show that each polynomial has just three distinct real roots. Let A be the largest root of p(x) and B the largest root of q(x). Show that $A^2$ + 3 $B^2$ = 4.