(ITA) Double Polynomial

Consider the polynomial $P(m) + 3m + 18 = \alpha m$ , where $\alpha$ is a real constant such that the sum of all roots of $P$ equals $3$ .

Evaluate the root $m$ such that two, and only two, roots of the polynomial $Q(n) = n^3 +m^{}n^2 + (m+4)n + 5 = 0$ are in the open interval $(-2, 2)$ .

This problem was adapted from ITA's 2013 Math Paper.