Let $q: \mathbb{R^3}\to \mathbb{R},\ q(x_1, x_2, x_3)=-5x_1^2-x_2^2-x_3^2+2x_1x_3+2x_2x_3-4x_1x_2$ be a quadratic form.
Find a maximal subspace $W \subseteq \mathbb{R^3}$ such that $\forall w \in W,\ q(w) \geq 0$.
Using row/column operations on $[q]_e$, I got to the canonical form: $\text{diag}(-1, -1, 1)$.
That tells me that there exists a basis $(u)=(u_1, u_2, u_3)$ such that: $[v]_u=\begin{pmatrix} x \\ y \\ z \end{pmatrix},\ q(v)=-x^2-y^2+z^2$
So if $w=\alpha u_3,\ q(w)=\alpha^2 \geq 0$, therefore $w \in W$. So far I've shown $\text{span}(u_3) \subseteq W$.
If we assume that there exists W\subset {W}' such that {W}' is maximal, then \forall {w}' \in {W}': q({w}')=-w_1^2-w_2^2+w_3^2\geq 0,\ [{w}']_u=\begin{pmatrix} w_1 \\ w_2 \\ w_3 \end{pmatrix}
But if we assume {w}' \notin W then $w_1 \neq 0$ or $w_2 \neq 0$.
At this point I'm stuck, not sure how to finish this up.