Suppose I have the following polynomials:
\begin{align} a(x) &= a_0+a_1x + a_2x^2 + \cdots a_nx^n \\ b(x) &= b_0+b_1x + b_2x^2 + \cdots b_nx^n \\ c(x) &= c_0+c_1x + c_2x^2 + \cdots c_nx^n \end{align}
where all of the coefficients for $a(x)$, $b(x)$, and $c(x)$ are real. The roots of these polynomials can be complex or real. Now suppose it is stated that the roots of the function:
$f(x) = a(x)b(x) + c(x)$
have negative real parts. Additionally, the roots of $a(x)$ and $c(x)$ also have negative real parts. Can any conclusions be made on the roots of $b(x)$?
Note: We are not given the actual values of the coefficients of the above polynomials.
Your help is much appreciated. Thanks!