I know I can suppose it is not irreducible and use Gauss' lemma to factorise $17715x^2 + 1234567x + 4561$ into two linear factors in $\mathbb{Z}[x]$ and try all the possible combinations (4561 is prime) to show that there is a contradiction, but it still looks clumsy. Is there any 'clean' approach? Please give me some hints, thank you.