Diophantine Equations of Degree 2

Question

During my studies, I have seen equations of this form $$xy+bx+cy+d=0$$ Is there a way to find solutions of equations of this form or the number of solutions of equations of this form without factoring or checking within a range?

Answer 1

Hint

Assuming $b,c,d$ are given integers,

$$(b+y)(c+x)=bc-d$$

Or $$y=-\dfrac{bx+d}{c+x}=-b+?$$