BG-MO 2012 Problem 5

Let $Q(x)$ be a quadratic trinomial. Given that the function $P(x)=x^{2}Q(x)$ is increasing in the interval $(0,\infty )$ , prove that: $$P(x) + P(y) + P(z) > 0$$ for all real numbers $x,y,z$ such that $x+y+z>0$ and $xyz>0$ .