 Solution

Write $\mathbf{M}=M\mathbf{n}$ with $|\mathbf{n}|=1$, and let $P=n_x^2{n}_y^2+n_x^2{n}_z^2+n_y^2{n}_z^2$. Since the nonnegative numbers $n_i^2$ sum to one,The quartic coefficient is $b+cP$. If $c\ge 0$ it is at least $b>0$. If $c<0$, the given condition $2b+c>0$ implies $b+c/3>b/3>0$. For $T>T_c$ the quadratic coefficient $a(T-T_c)$ is also positive. Hence every nonzero $M$ raises the free energy:The stability argument must apply to all orientations, rather than checking just one cubic direction. The stated inequality is sufficient for that purpose, although the sharp quartic stability condition for negative $c$ is the weaker $b+c/3>0$.