fingers crossed i didnt stuff up latex...
this is how i did it, hopign its all right
you basically just set up 2 equations, one hwere x = 1, the other x = -1, add the two together and the nc1, nc3,...nCn-1's all cancel
\textup{We know that}\\ nC1 + nC2.2^{2} + ... + nCn^{2} = n(n+1).2^{n-2}...