We choose $I = [-1,1]$ and fix any very good extremal pair $(Q, R)$. We set $P = QR$. By contradiction, show that $|P(-1)| = \|P\|_I$.
To do this, one may choose a real number $\epsilon > 0$, introduce the segment $I_\epsilon = [-1-\epsilon, 1]$ and bound the quantity:
$$\frac{\|Q\|_{I_\epsilon} \|R\|_{I_\epsilon}}{\|P\|_{I_\epsilon}}$$
using question 4.37.