How to solve Quartic Diophantine Equations

Question

I have this diophantine equation:$$x^4 - 4 x^2 y - 66 x^2 + 1089 = 0$$Is there a way to find all integer $x$ values that make $y$ also an integer without factoring or checking $x$ values within a range?

Answer 1

$$ x^4 - 4 x^2 y - 66 x^2 + 1089 = 0 $$ $$ x^4 - 2 (2 y + 33) x^2 + 1089 = 0 $$ $$ x^2 = 2 y + 33 +/- sqrt(4 y^2 + 132 y + 33^3 - 1089) $$ $$ x^2 = 2 y + 33 +/- sqrt(4 y^2 + 132 y) $$ $$ 4 y^2 + 132 y = M^2 $$ $$ (2y)^2 + 2 (2y) 33 - M^2 = 0 $$ $$ 2 y = - 33 +/- sqrt(1089 + M^2) $$ $$ 1089 + M^2 = W^2 $$ $$ (W-M) (W + M) = 1089 $$ $$ ... $$