Existence of solutions to $a^2x^2 + bx = y^2$ Is there a way to determine if there are or aren't any integer positive solutions $(x,y)$ to the equation $a^2x^2 + bx = y^2$ depending on the values of $a$ and $b$?

I tried to deal with it using Pell equations but I just couldn't work it out. Any help is appreciated.
Above equation shown below has solution:

$a^2x^2 + bx = y^2$

$x=3(k-3)^2$

$y=3(k-3)(3k-4)$

$a=2$

$b=15(k^2-4)$

For $k=5$    we get:

$(x,y,a,b)= (12,66,2,315)$
Above equation shown below has solution:

$a^2x^2 + bx = y^2$

$x=3(k-3)^2$

$y=3(k-3)(3k-4)$

$a=2$

$b=15(k^2-4)$

For $k=5$    we get:

$(x,y,a,b)= (12,66,2,315)$