What restrictions apply to the scalars $\alpha$ and $\beta$ in order for $$ \mathrm{tr}({\bf X}\cdot{\bf X}) + \alpha({\bf v}\cdot{\bf X}\cdot{\bf v})^2 + \beta({\bf v}\cdot{\bf X}\cdot{\bf X}\cdot{\bf v}) \ge 0 $$ to be fulfilled for any symmetric 3x3 matrix ${\bf X}$ and any normalized 3x1 vector ${\bf v}$?
Of course, the $\mathrm{tr}({\bf X}\cdot{\bf X})$ and $({\bf v}\cdot{\bf X}\cdot{\bf v})^2$ are strictly positive, and I suspect that ${\bf v}\cdot{\bf X}\cdot{\bf X}\cdot{\bf v}$ is too. In this case, it would seem that $\alpha,\beta\ge0$ is sufficient.
However, does a more general relationship exist between $\alpha$ and $\beta$ that fulfills the above inequality?