For a special case (when $a=\pm 1$) see 

http://mathoverflow.net/questions/56579/about-irreducible-trinomials

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.][1]


  [1]: http://wwwmaths.anu.edu.au/~brent/pd/QMt4.pdf