Another!!!

Hi all! This one is giving me trouble as well:

Show by mathematical induction that if n is a positive integer, then

(2*n)! < 2^2n*(n!)^2

Here's what I've done so far:

In proving the (n+1)th case I've done this:

(2(n+1))! = (2n+2)!

= (2n+2)*(2n+1)*(2n)!

< (2n+2)*(2n+1)*2^2n(n!)^2 (Using the inductive hypothesis)

< . . .

Now what? I need to get to < [ 2^(2n+2) ] * ((n+1)!)^2 in order for this proof to work.

Any help will be appreciated. Thanks!