Past exam of the mathematics course of the University of Cambridge / 2018 / ib / Paper 3 / 11G / c / Solution

Translate the variable by one:This satisfies Eisenstein criterion with $p=5$: every nonleading coefficient is divisible by $5$, while $25\nmid 30$. Translation $X\mapsto X+1$ is an automorphism of a polynomial ring, so it preserves irreducibility. Hence $f$ is irreducible over $\mathbb{Q}$.