Given integer constants $a,b,d$ ($a,b$ are coprime) and an integer variable $x$, is there a way to easily find for which $x$ does $gcd(a^2-db^2,a-bx)=a^2-db^2$ without substituting the values of $x$ individually and calculating the gcd? This is a continuation oy my gcd studies following the link Relation between two gcds.
Thanks