OFM 2022 S1

find all functions $f:\mathbb{Z} \to \mathbb{Z}$ such that $f(m+n)+f(m)f(n)=n^2(f(m)+1)+m^2(f(n)+1)+mn(2-mn)$ holds for all $m,n \in \mathbb{Z}$

Dataset text AI-MO AoPS. Quoted with source.

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.