How to get rid of the term with $xy$?

Question

I'm trying to put this conic on an identifiable form.

$$4x^2-4xy+y^2+20x+40y=0$$

I know that the term $xy$ implies that I need to rotate the conic so that $xy$ vanishes. But I've read on some books but I couldn't figure out how to do it. It seems that there is a system that needs to be solved, and this system involves some trigonometric funcions.

I thought about the following: As the term with $xy$ is going to be eliminated, I guess I should write:

$$4x^2-4xy+y^2+20x+40y=0\\ 4\left(x+\cfrac{5}{2}\right)^2+(y+20)^2-4xy-90=0$$

Perhaps the center of this conic is $(-5/2,-20)$. I guess that knowing the center must be important to something.

Answer 1

Hint The first three terms can together be rewritten as $$(2 x - y)^2,$$ suggesting part of a change of variable that will eliminate the cross-term.

Answer 2

Putting $x=x'-\frac{g+hy}a$ will not only remove xy term, but will also make the process easier.All the best.