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...

Let the factors $u:=y+x$ and $v:=y-x$ appear.
$$uv=\frac{9u-5v}2+4.$$
Then,
$$\left(2u+5\right)\left(2v-9\right)=61.$$

Update to please Misha Lavro:
The factorization of the LHS hints the introduction of these intermediate variables: as the RHS is linear, we know that we can transform to $uv=au+bv+c$, which factors as $(u-p)(u-q)=r$.
