AUSTRIAN-POLISH 1991 Problem 4

Let $P(x)$ be a real polynomial with $P(x) \ge 0$ for $0 \le x \le 1$ . Show that there exist polynomials $P_i (x) (i = 0, 1,2)$ with $P_i (x) \ge 0$ for all real x such that $P (x) = P_0 (x) + xP_1 (x)( 1- x)P_2 (x)$ .