Sign of the real parts of the solutions of $\det (s^2+Bs+C)=0$ where $B$, $C$ are real positive definite matrices. Let $ B, C$ be $n \times n$ symmetric, positive definite matrices with real entries and $I_n$ be the $n\times n$ identity matrix. Here, Consider the following polynomial equation with complex variable $s$. 
$\det ( s^2I_n + sB+C) = 0$
Can I conclude that all the $2n$ solutions (which is of course, counted with multiplicity) has  strictly negative real parts?
Note: The above conjecture is predicted by a physical argument, since the above equation comes out during an attempt to solve the equations of motion of an $n$- dimensional oscillator with some frictional force, by subsituting $x(t)=C e^{st}$. The existence of frictional force must cause the position to decay, which exactly amounts to $\Re \{{s\}} <0$. (This derivation is from Landau's Course of Theoretical Physics Vol. I, section 25.)
 A: Note first that the matrices $B$ and $C$ can be viewed as Hermitian matrices with positive real eigenvalues. 
Suppose that $\det ( s^2I_n + sB+C) = 0$ for some $s\in\mathbb{C}$, then there is a complex unit vector $\bf{x}$ such that $(s^2 I_n + sB + C) \bf{x} = \bf{0}$. Then it follows by multiplying $\bf{\overline{x}^T}$ to the left that 
$$
s^2 + s \beta+\gamma=0 \ \   
$$
where $\beta=\bf{\overline{x}^T}\it B \bf{x}$, $\gamma=\bf{\overline{x}^T}\it C\bf{x}$.
Since $B$ and $C$ are positive definite, we have $\beta>0$, $\gamma>0$. 
Then, such number $s\in\mathbb{C}$ is a root of the equation 
$$
z^2 + \beta z + \gamma=0 \ \ \ (*).
$$
This equation has either one of two possibilities for the location of roots:
Case 1 (Both roots are not real: $\beta^2-4\gamma<0$)
By the quadratic formula, the real parts of the roots are $-\beta/2$. 
Case 2 (It has real roots: $\beta^2-4\gamma\geq 0$)
We cannot have nonnegative real roots since $\beta>0$, $\gamma>0$. Thus, any real roots must be negative. 
Therefore, any roots of the equation
$$
\det(s^2 I_n + sB + C)=0
$$
must have negative real parts. 

It would be interesting to know how to derive such equation from an attempt to solve the equations of motion of an $n$- dimensional oscillator with some frictional force. It would be great if you can add the derivation in your question, or point us to a reference. 
