FFT for expanded form of equation multiplication
The trivial algorithm that multiplies every monomial in $A$ by every monomial in $B$ takes time $O(|A| \cdot |B|)$ (where $|A|$ is the number of monomials in $A$ or $\deg A + 1$, depending on the representation), which is the same order of magnitude as the size of the output, and so optimal. You only need FFT if you want to actually compute the product $AB$. In particular, there is no way to compute your function in time $O(n\log n)$, simply because the length of the output is $\Omega(n^2)$.