Question 5
Given two arithmetic progressions $(a_n)$, $(b_n)$ and an integer $m \gt 2$. Consider $m$ quadratics $P_k(x) = x^2 + a_k x + b_k$, $k = 1, 2, \dots, m$. Suppose that neither $P_1(x)$ nor $P_m(x)$ has a real root. Prove that none of the given quadratics contains a real root.