We choose $I = [-1,1]$ and write $C_{n,m}$ instead of $C_{n,m}^I$. Show that there exists a polynomial $Q_4$ whose roots are all in $I$ and such that the pair $(Q_4, R_2)$ forms a good extremal pair.
To do this, given a root $w$ of $Q_3$ that is not in $I$, one may introduce the polynomial:
$$S_3(X) = \frac{X-1}{X-w} Q_3(X)$$
then one may follow the method used in the two previous questions.