Question 2
Let $(a_n)$ and $(b_n)$ be arithmetic progressions, and let $m\gt 2$ be an integer. Consider $P_k(x)=x^2+a_kx+b_k$, $k=1,2,\ldots,m$. Prove that if $P_1$ and $P_m$ have no real roots, then none of the remaining polynomials has a real root.