Given this equation $x^2+y^2+z^2+2xy+2xz+2yz-x-y-z=6$ and the corresponding quadric:
If I rearrange the equation to $(x+y+z-3)(x+y+z+2)=0$ (which is equivalent), I get:
So, which is the right quadric to my equation? Why does it have two representations?
Just as in your question How to transform (rotate) this hyperbola?, it appears you've omitted the multiplication symbol between the parenthesized factors in the second plot.