We choose $I = [-1,1]$ and fix any very good extremal pair $(Q, R)$. We set $P = QR$ and denote by $x_1 \leq \ldots \leq x_{n+m}$ the roots of $P$ counted with multiplicity. Show that:
$$Q = \prod_{k=m+1}^{n+m}\left(X - x_k\right) \quad \text{and} \quad R = \prod_{k=1}^{m}\left(X - x_k\right)$$