Eisenstein's criterion

Often Eisenstein's criterion does not apply for any prime number.

This property is also useful when combined with properties such as Eisenstein's criterion.

(ii) Eisenstein's criterion shows that there exist in Q[x] infinitely many prime (i.e. irreducible) polynomials of each degree.

Notably he obtained Hensel's lemma before Hensel, Scholz's reciprocity law before Scholz, and formulated Eisenstein's criterion before Eisenstein.

In this particular example it would have been simpler to argue that H (being monic of degree 2) could only be reducible if it had an integer root, which it obviously does not; however the general principle of trying substitutions in order to make Eisenstein's criterion apply is a useful way to broaden its scope.