foil nightmare within mods

Alright, the problem goes as follows

in mod 5, compute (a+b)^5

in mod 3 compute (a+b)^3

in mod 2 compute (a+b)^2

in mod 4 compute (a+b)^4

in mod 7 compute (a+b)^7

based on results of your findings, what do you thing (a+b)^n is equal to in mod n? Explain in detail.

Now... the a's and the b's represent classes with in the mods, and I've solved all of them out to my satisfaction except mod 4.

i have a^5 + b^5 in mod 5

a^3 + b^3 in mod 3

a^2 + b^2 in mod 2

a^7 + b^7 in mod 7

however, in mod 4, i have a^4 + 6(a^2)(b^2) + a^4

I've tried solving it out several ways, and I can't make mod 4 as pretty as the other mods. Why is this....? Am I doing the foil wrong? because I've tried it several ways to see if I was missing something, and almost thought i had it until i noticed I made a mistake and then got the result that is making it soooo not nice and neat as the others. Please help (Doh)