An Olympiad question about equality How many $(x, y)$ positive integer pairs have $y^2-x^2=2y+7x+4$ equality?
I can't solve this Olympiad question.
$\textbf{Solution:}$ If above equality is regulated,we obtain $(2x+2y-5)(2x-2y+9)=29$. And, $29$ is prime we can easily find solution.



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*$\textbf{1-)}$ How can we regulate it to obtain $(2x+2y-5)(2x-2y+9)=29$?


*$\textbf{2-)}$ Are there the general method for this kind of questions?


Thank you for help...

It's just completing the square,
$$y^2-2y-(x^2+7x)=4$$
$$y^2-2y+1-1-\left(x^2+7x + \left(\frac72\right)^2 \right)+\left(\frac72\right)^2=4$$
$$(y-1)^2-1-\left(x+ \frac72 \right)^2+\left(\frac72\right)^2=4$$
Multiply everything by $4$.
$$(2y-2)^2-4-\left(2x+ 7 \right)^2+49=16$$
