I have $N$ monic polynomials of degree $(N-1)$, with complex unknown $x\in \mathbb{C}$: $\text{Pol}_{i}(x)$.
I take a linear combination of those polynomials, obtaining another polynomial of degree $(N-1)$: $\text{Pol}_{sum}(x)=\sum_{i}a_{i}\text{Pol}_{i}(x)$, with complex unknown $x\in \mathbb{C}$. I want to show that exists at least one linear combination with real coefficients $a_{i}\in \mathbb{R}$ such that $\text{Pol}_{sum}(x)$ has no roots with positive real part.
P.S. I feel that the answer is positive and it has to do with the vector space of the polynomials of degree $(N-1)$ or less.