assuming $S_n$ is True.

$2^{n+1} = 2\cdot 2^n > 2n^2$

$(n+1)^2 = n^2 + 2n + 1$

is $2n^2 > n^2 + 2n + 1$ ?

$n^2 \overset{?}{>} 2n + 1$

If you plot it you'll see this is true for $n>5$

It's a simpler induction problem to prove it.

If you can prove this you'll have shown that

$2^{n+1} > (n+1)^2$ which is what you need to show for the original induction problem.