How to find $x$ and $y$ in $(x+c_1)(y+c_2)=c_3$ where $x, y \in \mathbb{N}$ and $c_1,c_2,c_3$ are non-zero constants?

Question

Is there a general formula to find $x$ and $y$ in $(x+c_1)(y+c_2)=c_3$ where $x, y \in \mathbb{N}$ and $c_1,c_2,c_3$ are non-zero constants?

Because I was struggling trying to find solutions to this equation: (Picking two positive integer and trying) $$(x+1)(y+3)=35$$

Answer 1

$$ (x+c_1)(y + c_2) = c_3 \Rightarrow x + c_1= \frac{c_3}{y+c_2} \Rightarrow x = \frac{c_3 - c_1y-c_1c_2}{y + c_2 } $$

$$ y + c_2 \mid c_3 - c_1(y+c_2) \Rightarrow y + c_2 \mid c_3 $$

so find the factors of $c_3$ and you'll find $ x $ and $ y $