We prove that if
$g\left(x,y\right)$ is a
polynomial of degree
$d$ that
is not a polynomial of only
$y$,
then for any finite set
$A\subset \mathbb{R}$
$$\leftX\right{\gg}_{d}A{}^{2},\phantom{\rule{1em}{0ex}}\text{where}X:=\left\{\frac{g\left({a}_{1},{b}_{1}\right)g\left({a}_{2},{b}_{2}\right)}{{b}_{2}{b}_{1}}:{a}_{1},{a}_{2},{b}_{1},{b}_{2}\in A\right\}.$$
We will see this bound is also tight for some polynomial
$g\left(x,y\right)$.
