Consider the quadratic form $q(x,y,z)=4x^2+y^2−z^2+4xy−2xz−yz $ over $\mathbb{R}$ then which of the following are true Consider the quadratic form $q(x,y,z)=4x^2+y^2-z^2+4xy-2xz-yz$ over $\mathbb{R}$. Then which of the followings are true?
1.range of $q$ contains $[1,\infty)$
2.range of $q$ is contained in $[0,\infty)$
3. range=$\mathbb{R}$
4.range is contained in $[-N, ∞)$ for some large natural number $N$ depending on $q$
I am completely stuck on it. How should I solve this problem?

If you consider that for $x=0$ and $y=0$ we have that $q$ maps onto $(-∞,0]$ because $q(0,0,z)=-z^2$, and for $x=0$,$z=0$ we have that $q$ maps onto $[0,∞)$, then as a whole $q$ maps onto $(-∞,∞) = \mathbb{R}$.
