I am looking for "fast",pencil and paper technique for factoring a bi-variate quadratic polynomial, assuming that the polynomial is of the form $$ax^2 + bxy + cy^2 + gx + fy + d,$$
where $a,b,c,g,f,d \in \mathbb{N_0}$.
Please explain with an example.
Let $y=0$, and factor the resulting one-variable quadratic (if possible) as $(rx+s)(r'x+s')$. Let $x=0$, and factor as $(ty+u)(t'y+u')$. Check to see whether your two factorizations are compatible. If so, they give you the factorization of the original; if not, there isn't one.