The first step for this problem is to map out what the probability that x is even would be:
$ P[X is Even]= \sum\limits_{k=0,even}^N P[x=k]= \sum\limits_{k=0}^N {N \choose k} P^k(1-P)^{N-k}[1+(-1)^k] $
Next we must expand $ (x+y)^n + (x-y)^n $ using the binomial thereom:
$ (x+y)^n + (x-y)^n = \sum\limits_{k=0}^N {N \choose k} y^kx^{N-k} + \sum\limits_{k=0}^N {N \choose k} (-y)^kx^{N-k} $
This simplifies to:
$ \sum\limits_{k=0}^N {N \choose k} (x)^{N-k}y^k[1+(-1)^k] $
If you compare P{X is Even} with the binomial expansion just above then you can derive a condensed equation in the form of $ (x+y)^n + (x-y)^n $ where $ x = 1-P $ and $ y=P $:
$ ((1-P)+P)^n + ((1-P)-P)^n = 1^n + (1-2P)^n $
