Proof that language is not context-free with Parikh's theorem
If $L$ were context-free then, applying the homomorphism mapping both letters to $a$, also the following language would be context-free:
$$
L' = \{ a^{2n^2+2n} : n \geq 0 \}.
$$
According to Parikh's theorem, a unary language is context-free iff the corresponding subset of $\mathbb{N}$ is eventually periodic. Yet the sequence $2n^2+2n$ has arbitrarily long gaps, since $2(n+1)^2+2(n+1)-2n^2-2n = 4n+4$. Therefore the set $\{2n^2+2n : n \geq 0\}$ is not eventually periodic, and so $L'$ is not regular or context-free.