I have a question that I have worked out for the most part, but there is one small component of the answer that is confusing me.

The question:

Identify the following conic. That is, is it a circle, parabola, hyperbola or ellipse? Show why.

Working:

Completing the square for x and y nets and ,respectively.

Therefore,

My issue lies in factoring my revised equations, I can do it just fine but the answers don't appear to match up, when I use F.O.I.L on I get (x^2-2x-2x-4); when I do the same operation on I get: which looks like the right answer to me....however, isn't a square.

I have the same problem with the y side of the equation as well.

I think I might be doing something wrong or making a mistake along the way but I can't see what it is.

What am I not getting?