VN-TST 2009 Problem 2

Let a polynomial $P(x) = rx^3 + qx^2 + px + 1$ $(r > 0)$ such that the equation $P(x) = 0$ has only one real root. A sequence $(a_n)$ is defined by $a_0 = 1, a_1 = - p, a_2 = p^2 - q, a_{n + 3} = - pa_{n + 2} - qa_{n + 1} - ra_n$ . Prove that $(a_n)$ contains an infinite number of nagetive real numbers.