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:
a^{2} - b^{2} = (a + b)(a - b)
and this one, too, is fairly common:
a^{3} - b^{3} = (a^{2} + ab + b^{2})(a - b).
it's not too hard to see that:
a^{4} - b^{4} = (a^{2} + b^{2})(a + b)(a - b) = (a^{3} + a^{2}b + ab^{2} + b^{3})(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.