A family of four-variable expanders with quadratic growth

We prove that if $g(x,y)$ is a polynomial of constant degree $d$ that $y_2-y_1$ does not divide $g(x_1,y_1)-g(x_2,y_2)$, then for any finite set $A \subset $\mathbb{R}$$ $$\mathrm{\mid }X\mathrm{\mid }{\gg }_d\mathrm{\mid }A{\mathrm{\mid }}^2,\text{where }X:=\{\frac{g(a_1,b_1)-g(a_2,b_2)}{b_2-b_1}:a_1,a_2,b_1,b_2\in A\}\mathrm{.}$$ We will see this bound is also tight for some polynomial $g(x,y)$.