I am trying to show that the following is true:
Suppose $f(x)$, monic in $\mathbb{Z}[x]$, factors modulo 3 into the product of two irreducible polynomials of degree 2, and factors modulo 2 into the product of an irreducible polynomial of degree 3 and a polynomial of degree 1. Show that $f(x)$ is irreducible in $\mathbb{Q}[x]$.
I've tried writing down the possible factors of the given degrees mod 2 and 3 and then trying to see if those lead to coefficients of $f(x)$ such that the Eisenstein criterion can be applied to show irreducibility, but this hasn't really seemed to work. Not quite sure how else to approach this, so I'd appreciate any help. Thanks!