= Solution
Brouwer's fixed-point theorem in the plane states that every continuous map from a closed disk to itself has a fixed point. Equivalently, the same holds for every nonempty compact convex subset of $\mathbb R^2$.
On the closed unit disk define
$$
F(z)=-\frac{z^6-2z^5+4z^2+2}{9}.
$$
For $|z|\leq1$,
$$
|F(z)|\leq\frac{1+2+4+2}{9}=1,
$$
so $F$ maps the disk continuously into itself. Brouwer supplies $z$ with $F(z)=z$, which is equivalent to
$$
\boxed{z^6-2z^5+4z^2+9z+2=0.}
$$
\b[Thus this <polynomial> has a root with $|z|\leq1$.]
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