  We show that if $h(x,y)=ax^2+bxy+cy^2\in \mathbb{Z}[x,y]$ satisfies
$\Delta(h)=b^2-4ac\neq 0$, then any subset of $\{1,2,\dots,N\}$ lacking nonzero
differences in the image of $h$ has size at most a constant depending on $h$
times $N\exp(-c\sqrt{\log N})$, where $c=c(h)>0$. We achieve this goal by
adapting an $L^2$ density increment strategy previously used to establish
analogous results for sums of one or more single-variable polynomials. Our
exposition is thorough and self-contained, in order to serve as an accessible
gateway for readers who are unfamiliar with previous implementations of these
techniques.
