I'm trying to figure out how I can do this for some arbitrary function. Say I find a monic associate of $f$ that we'll call $f_1(x)$. If I then apply Eisenstein's Criterion or Descartes' Rational Root Test to $f_1(x)$ and find it's irreducible, what does this say about $f(x)$? Is there a theorem for this?
Hint $\rm\ f\:$ reducible $\Rightarrow$ $\rm\:uf\:$ reducible, since $\rm\:f = gh\:$ $\Rightarrow$ $\rm\:uf = (ug)h.\:$ Thus, contrapositively, we deduce that $\rm\:uf\:$ irreducible $\rm\Rightarrow\: f\:$ irreducible (here $\rm\:u\:$ is any unit).