Math Help - Prime polynomial

1. Prime polynomial

Prove that there is no non-constant polynomial f(x), with integer coefficients, such that f(x) is prime for all integers x.

I just started modular arithmetic, so if someone can explain to me how to apply modular arithmetic to prove this, it would be much appreciated!

2. No polynomial $f (n)$ with integral coefficients, not a constant, can be prime for all $n$, or for all sufficiently large $n$.
We may assume that the leading coefficient in $f(n)$ is positive, so that $f(n)\rightarrow\infty$ when $n\rightarrow\infty$, and $f(n) > 1$ for $n > N$, say. If $x > N$ and $f(x)=a_0x^k+\ldots =y>1$ then $f(ry+x)=a_0\left(ry+x\right)^k+\ldots$ is divisible by $y$ for every integral $r$; and $f(ry+x)$ tends to infinity with $r$. Hence there are infinitely many composite values of $f(n)$.