How much easier is testing polynomials of the form x^n + ax + b for irreducibility (in Z[x]) than testing polynomials in general? I am especially interested in the case where n is prime, which may be easier than the case of arbitrary n.

For a special case (when $a=\pm 1$) see Testing irreducibilty over small moduli (obviously, if your polynomial is irreducible modulo $p,$ it is irreducible), is described in detail in this paper of Richard Brent's. 

