$\sum_{k=0}^{n} {n \choose k}^2 (1+x)^k (1-x)^{n-k}$ as a function of $(1-x^2)$

Question

I am trying to show that $$\sum_{k=0}^{n} {n \choose k}^2 (1+x)^k (1-x)^{n-k}$$ can be expressed explicitly as a function of $(1-x^2)$.

I am wondering where to start with this, as anything I have tried has not gone very far!

Thank you!

Answer 1

Hint:

If you assume $k<\frac n2$, then $n-k>k$ and you have a summand of $\binom nk^2(1-x^2)^k(1-x)^{n-2k}$

If $k=\frac n2$ you just get $\binom{n}{\frac n2}^2(1-x^2)^{\frac n2}$

What about when $k>\frac n2$?