TUYMAADA 2012 S2

Let $P(x)$ be a real quadratic trinomial, so that for all $x\in \mathbb{R}$ the inequality $P(x^3+x)\geq P(x^2+1)$ holds. Find the sum of the roots of $P(x)$ .

Proposed by A. Golovanov, M. Ivanov, K. Kokhas