Proving an Inequality Involving Integer Partitions

Proving an Inequality Involving Integer Partitions

I am having a bit of trouble beginning the following:
Prove that for all positive integers $n$, the inequality
$p(n)^2<p(n^2+2n)$ holds, where $p(n)$ is defined as the number of all
partitions of $n$.
I initially considered weak induction on n, but am not sure if that is the
correct way to begin. Is there an alternate, stronger path (such as a
combinatorial proof) I should consider? I feel like I'm making this more
difficult than it should be, and I apologize if this is the case.
Thank you in advance!