Proving a polynomial $f(x)$ composite for infinitely many $x$ Let $f(x)=a_0+a_1x+ \ldots +a_nx^n$ be a polynomial with integer
  coefficients, where $a_n&gt;0$ and $n \ge 1$. Prove that $f(x)$ is
  composite for infinitely many integers $x$.


I can easily show that there are infinitely many composite numbers of the form $a_0+a_1x+ \ldots +a_nx^n$ if $a_0 \ge 2$, we just note that $f(x)$ is composite for every $x$ being a multiple of $a_0$. But I can't find a way to prove this in the case $a_0=1$.
Choose $m$ such that $f(m)\ne\pm1$, then choose any prime $p$ dividing $f(m)$, and think about $f(m+pk)$ for $k=1,2,\dots$.
Let $x_k=(\text{lcm}(a_0,a_1,\dots,a_n))^k$ for some $k$. Then all $x_k$ with $k\in\mathbb N$ are values which produce composite numbers.