Question: Identify the type of surface given by $x^2 + 2xy + z^2 = 1$.
Solution: By completing the square, one obtains $x^2 + 2xy + z^2 = 1$, so $(x + y )^2 - y^2 + z^2 = 1$, and then $(x + y)^2 + z^2 = 1 + y^2$. It follows that the surface is a hyperboloid of one sheet.
My doubt is: Why did he conclude that the surface is a hyperboloid of one sheet, just completing the square ?
The various Quadric Surfaces have nice characterizations, all of which are listed on Wikipedia:
https://en.wikipedia.org/wiki/Quadric#Euclidean_space