I answered your first question in the comments. To answer your second question, yes, these general principles go under the name of "Hensel lifting", see for example these <A HREF="http://people.csail.mit.edu/madhu/FT98/lect7.ps">lecture notes</A>.

For a computer algebra implementation, you could try <A HREF="http://www.sagemath.org/doc/constructions/polynomials.html">Sage</A>. I tried it myself, and it cannot find any nontrivial factorization of your polynomial.