Prove $f$ entire and with a pole at infinity to be polynomial

Well, $g(0)=f\left(\frac10\right)=f(\infty)=\infty$.

Note that when $z\rightarrow \infty, w=\frac{1}{z}\rightarrow 0$. And since $f(z)\rightarrow \infty$ as $z\rightarrow \infty$, there is no essential singularity. So $g(w)$ must have a pole of finite order at $\infty$.