Let $V := M_2(\mathbb{R})$ and let function $f : V \times V \to \mathbb{R}$ be defined by
$$f \left( A, B \right) = 2 \mbox{tr} \left( A B \right) -\mbox{tr} \left( A \right) \mbox{tr} \left( B \right)$$
I showed that it is a bilinear form on $V$ and I found that the rank of its quadratic form $q$ is $9$, and the sign is $(9,0)$.
The next exercise is to find a subspace of $V$ of maximal dimension that $q|_W$ is nondegenerate.