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This question already has an answer here:

Show there doesn’t exist a non-constant polynomial $p(x)$ with integer coefficients such that $p(x)$ is prime for all non-negative integers $n$.

There is a hint: note in particular that $p(0)$, the constant term of $p(x)$, is prime.

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marked as duplicate by steven gregory, darij grinberg, Claude Leibovici, mechanodroid, Henrik Sep 26 '17 at 16:01

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    $\begingroup$ Once you know that the constant $q = p(0)$ is a prime, or perhaps $-q$ is the prime, what happens to the value of the polynomial when $x$ is divisible by $q?$ $\endgroup$ – Will Jagy Sep 25 '17 at 1:28
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    $\begingroup$ Suppose that $p(0)=a\in\Bbb P$. Notice that $p(a)$ is composite iff $p$ is a non-constant polynomial satisfying the constraints in the problem. $\endgroup$ – Prasun Biswas Sep 25 '17 at 1:29
  • $\begingroup$ Try $x^2 + x + 5.$ What happens when $x$ is divisible by $5?$ $\endgroup$ – Will Jagy Sep 25 '17 at 1:43