$p(x)\geq 0 \forall x\Rightarrow p(x)+p'(x)+p''(x)+...+p^{(n)}(x)\geq 0$ $p(x)\geq 0 \forall x \in \mathbb{R} \Rightarrow p(x)+p'(x)+p''(x)+...+p^{(n)}(x)\geq 0$, where p(x) is a polynomial of degree n.
I showed:
$a_{n}+...+a_{0}\geq 0$,
$p(x)+p'(x)+p''(x)+...+p^{(n)}(x)=\sum_{m=0}^{n}\sum_{k=m}^{n}a_{k}\frac{n!}{(n-k)!}p(x)^{(k-m)}=\sum_{k=0}^{n}\sum_{m=0}^{k}a_{k}\frac{n!}{(n-k)!}p(x)^{(k-m)}$
 A: If $n$ is odd then $p(x)$ becomes negative as $x\to\infty$ or $x\to-\infty$. Hence we may assume $n$ is even.
The function $f(x)= p(x)+\ldots +p^{(n)}(x)$ is also a polynomial of degree $n$ and hence assumes its global minimum at some $a\in\mathbb R$. Then $f'(a)=0$. But from $p^{(n+1)}(x)=0$ we see $f'(x)=f(x)-p(x)$ and hence $f(a)=p(a)\ge 0$. As $a$ is the global minimum, $f(x)\ge f(a)\ge 0$ for all $x$.
A: Notice
$$\sum_{k=0}^n p^{(k)}(x) = \sum_{k=0}^n p^{(k)}(x)\int_0^\infty \frac{t^k}{k!} e^{-t}dt
= \int_0^\infty \left(\sum_{k=0}^n \frac{p^{(k)}(x)}{k!}t^k\right) e^{-t}dt\\ 
= \int_0^\infty p(x+t) e^{-t} dt
$$
If $p(x) \ge 0$ for all $x$, then $\sum\limits_{k=0}^n p^{(k)}(x)$ is the integral over a non-negative function and hence is non-negative itself.
A: An another proof,
$Q=P+P'+...+P^{(n)}$
Notice 
$$
P=Q-Q'
$$
Thus 
$$
\forall x\in \mathbb{R}, Q(x)\geq Q'(x).
$$
Let
$$
 \forall x \in \mathbb{R}, \varphi(x)= e^{-x}Q(x)
$$
So,
$$
\varphi'(x)=e^{-x}(Q'(x)-Q(x))\leq 0
$$
Therefore $\varphi$ is decreasing on $\mathbb{R}$. She tends to $0$ when $x \longrightarrow +\infty$, so $Q(x)\geq 0$
