Given an irreducible polynomial of structure $$f(x,y)=\sum_{\substack{i,j\in\{0,1,2\}\\i+j\leq3}}a_{i, j}x^iy^j\in\mathbb Z[x,y]$$ with $a_{2,1}a_{1,2}a_{1,1}a_{1,0}a_{0,1}a_{0,0}\neq0$ if $f(m,n)=0$ with $m, n\in\mathbb Z$ then is it possible to decompose $f(x,y)$ as $$f_1(x,y)+g_1(x)h_2(y)+h_1(y)g_2(x)$$ where $f_1(x,y)$ is degree $1$ polynomial and $g_i(x)$ and $h_i(y)$ are degree $i\in\{1,2\}$ polynomials in $\mathbb Z[x]$ and $\mathbb Z[y]$ respectively where$g_1(m)=g_2(m)=0$ and $h_1(n)=h_2(n)=0$ still holds?
How difficult is to get such decompositions?
