There are some problems in high school Olympiad that ask to find integer solutions of the form $Q(x^2) = dy^2 (*)$ where $Q$ is a quadratic polynomial and $d$ is an absolute constant and quite often, $Q(0)= \pm 1$. Some times, $(*)$ can transform into finding all square elements in a Pell-equation's root sequence and look up the remainder for some modulus, and some times $(*)$ can also be solved by using infinite descent. However, none of those methods are reliable because the associated sequence of $(*)$ is periodic for every modulus so it can not be used if $(*)$ has solutions, and sometimes infinite descent is not possible. But $(*)$ is also equivalent to $dxQ(x) = y^2$ which is a typical elliptic curve equation and it is known to only have finitely many integer solutions (https://en.wikipedia.org/wiki/Siegel%27s_theorem_on_integral_points). So are there any reliable ways of cracking these types of equation in elementary (high-school level) way?