There's a whole class on factoring???
Anyway...
from the original problem it's a trinomial square, so that factors to (a^2-1)(a^2-1) hence a^4 - a^2 - a^2 + 1. Remember FOIL.
Those two are a difference of squares so you get (a+1)(a-1)(a+1)(a-1)
Pretty easy once you get used to it.