Irreducible in $\Bbb Q[x]$

Question

Suppose $f(x)$ is an polynomial of integer coefficients. If for infinitely many integers $x$, $f(x)$ is prime. Show that $f(x)$ is irreducible in $\Bbb Q[x]$.

Suppose $f(x)$ is is reducible in $\Bbb Q[x]$, then what...

Answer 1

Then $f(x)=g(x)h(x)$ where $\deg g,\deg h< \deg f$. If $f(x)$ is a prime for infinitely many integers $x$, then either $g(x)$ or $h(x)$ is $1$ for infinitely many $x$, so...