Basic multivariate Factorization type question I came across this polynomial $$ x^2 - 2kxy + ky^2 +d; \, k>0 $$ in some of my work and was wondering if there was a trick to coercing/factoring it into a polynomial of the form $$ (x-x_0)^2-(y-y_0)^2+r $$ for some $x_0,y_0,r \in \mathbb{R}$. I was thinking a completing the square type approach but embarrassingly enough I'm stumped. All help is appreciated. Thanks.

We can coerce the polynomial into a nicer form through $$x^2-2ky+k^2y^2-(k^2-k)y^2+d=(x-ky)^2-(k^2-k)y^2+d$$ And if we set $u=x-ky$, we have $$u^2+(k-k^2)y^2+d$$ The substitution is necessary, since as stated in the comments, we cannot write it as $(x-x_0)^2+(y-y_0)^2+r$ because of the xy term.