Assume that the leading coefficient is positive (other case is analogous). Since $P(x)\to \infty$ as $x\to\infty$ and $P(x)\to-\infty$ as $x\to -\infty$, we conclude from Intermediate Value Theorem that $P(x)$ is surjective. Next, let $r_1<\cdots<r_d$ be roots of $P$. From surjectivity, there exists distinct $x_i$ such that $P(x_i)=r_i$. But then, $P(P(x_i)) = P(r_i) = 0,\forall i$, so any $x_i$ is a root of $P\circ P$.