Let $P$ be a real polynomial of exact degree $2n$ ($n \geq 1$) whose zeros are real numbers and such that
\begin{equation*}
P(j) \geq 0
\quad \text{for any} \quad
j \in \mathbb{Z}.
\end{equation*}

Does there exist non-negative real numbers $\alpha_0,\alpha_1,\ldots,\alpha_n,$ with at least one of the $\alpha_i$ non-zero, such that the polynomial
\begin{equation*}
Q(x) = \sum_{k=0}^{n} \alpha_k P(x+k)
\end{equation*}
is non-negative on the whole real line, i.e.; $Q(x) \geq 0$ for any $x \in \mathbb{R}$ ?

I would like to add that this question is not merely a mathematical curiosity but pops up naturally while working on spectral transformations of discrete measures.