IN-TST 2010 Problem 2

Two polynomials $P(x)=x^4+ax^3+bx^2+cx+d$ and $Q(x)=x^2+px+q$ have real coefficients, and $I$ is an interval on the real line of length greater than $2$ . Suppose $P(x)$ and $Q(x)$ take negative values on $I$ , and they take non-negative values outside $I$ . Prove that there exists a real number $x_0$ such that $P(x_0)<Q(x_0)$ .