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1) Let $f(z)=e^{-z}+z^2-4z+4.$ Show that $f$ has exactly two zeroes on $\{z\in\mathbb C:|z-2|<1\}.$ Show that these zeroes are distinct, that is, it's not a zero of order two.

2) Let $f,g\in\mathcal H(\mathbb C)$ be no constant. Prove that $(f\circ g)(z)$ is a polynomial iff $f$ and $g$ are polynomials.

__Attempts__:

1) I know I have to use Rouché's Theorem, but I don't know how for this case, it annoys me the $|z-2|<1.$

2) The $\Longleftarrow$ implication is trivial, but not the another one. I don't know how to proceed though.