Are you asking how one can come up with this equation or how to prove it? To prove it, multiply through the right-hand side and almost all terms cancel out.
well, most people know this one:
a2 - b2 = (a + b)(a - b)
and this one, too, is fairly common:
a3 - b3 = (a2 + ab + b2)(a - b).
it's not too hard to see that:
a4 - b4 = (a2 + b2)(a + b)(a - b) = (a3 + a2b + ab2 + b3)(a - b)
do you see the pattern...? once you are at this point, it's natural to conjecture that:
, which you say you can prove.