= Solution
Put $\kappa=k/m$, $q=g/l$. The positive stiffness-per-mass <matrix> is
$$
K=\begin{pmatrix}\kappa+q&-q\\-q&q\end{pmatrix}.
$$
Substitution of a <normal mode> $x=v\cos\omega t$ requires $Kv=\omega^2v$. Its <characteristic equation> is $\lambda^2-(\kappa+2q)\lambda+\kappa q=0$. Therefore
$$
\boxed{\omega_\pm^2=\frac\kappa2+q\pm\sqrt{\frac{\kappa^2}{4}+q^2}.}
$$
For $k,m,g,l>0$, their sum and product are positive, so both squared <angular frequencies> are positive and $\omega_+>\omega_->0$.
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