As a bonus question on a Galois Theory worksheet, there is the question: "Prove that $f(x)=x^n+x+3$ is irreducible in $\mathbb{Q}[x]$ for $n>1$ "
I know that I can prove this by simply showing that it is irreducible in $\mathbb{Z}[x]$ and using Gauss's Lemma, but I cannot simply prove this irreducibility by applying Eisenstein's Criterion. Also, I cannot see how you can prove this statement by induction.
What are the best strategies to solve this problem?