So it's false for n=6. If it's false for n, can you prove that it's false for n+1? In other words, given n! > (n+1)2^n, can you prove (n+1)! > (n+2)2^(n+1)?
As a hint, since you have:
n^2 > 3
n^2+1 > 4
n^2+2n+1 > 2n+4
(n+1)^2 > 2(n+2)
So, by induction, it's false for all n>5. And you have the result for 1 <= n <= 5, so that accounts for all the natural numbers.
Post again in this thread if you're still having trouble.