Possibility of poles with positive real part in RLC filter
Do the math for \$s_{1,2}\$, looking at the numerator. Assuming no component is zero, the term \$\sqrt{L^2 - 4R^2 LC}\$ will have a real part no larger than \$L\$. Therefore, the real part of \$-L \pm \sqrt{L^2 - 4R^2 LC}\$ will have a range of \$(0, -L]\$. This gives the result your professor arrived at.

This analysis is valid for the expected linear, positive-valued devices implied by the question and your professor's response. This result does not necessarily hold for non-linear devices that exhibit negative resistances.