I've been thinking on this problem for 2 days but I haven't found even a clue. The question is to prove the correctness of the following identity:

$\displaystyle \displaystyle \sum _{k=0}^{p} {p\choose k}^{2}{n+2\,p-k\choose 2\,p}= {n+p\choose p} ^{2}$

Could you give me a hint about how to prove this?