KöMaL 2015 A656

Let $p(x)=a_0+a_1x+\dots+a_nx^n$ be a polynomial with real coefficients such that $p(x)\ge0$ for $x\ge0$. Prove that for every pair of positive numbers $c$ and $d$ , $a_0 + a_1(c+d) + a_2(c+d)(c+2d) + \dots + a_n(c+d)(c+2d)\dots(c+nd) \ge0$.