= Solution
<Rouché's theorem> states that if $f,g$ are <holomorphic functions> on a neighbourhood of a simple closed contour and its interior, and $|g|<|f|$ on the contour, then $f$ and $f+g$ have the same number of zeros inside, counted with <multiplicity>.
On $|z|=1$, use $f=6z^2$: the other terms have absolute value at most $1+3+1=5<6$. Hence there are two zeros inside the unit circle. On $|z|=2$, use $f=3z^7$: its modulus is $384$, whereas the other terms have total modulus at most $256+24+1=281$. Hence there are seven zeros inside the larger circle. The strict inequalities also exclude zeros on either boundary. \b[There are $7-2=5$ roots in the annulus, counted with multiplicity.]
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