 Solution

For $0<\theta <\pi /2$ and $\lambda >\sin 2\theta$, the discriminant is strictly positive. The two roots have positive sum $\lambda /c^2$ and positive product $s^2/c^2$, so both roots are positive and distinct. Each gives exactly one positive $\alpha$ and hence a different right stretch tensor. If $\lambda \ne 1$, neither tensor is the undeformed identity because its third diagonal entry is $\lambda$. Thus two distinct deformed configurations are possible. Excluding $\lambda =1$ removes the case in which one of the two configurations is the undeformed body; it is not needed just for positivity of the roots.