Show $z^4 + e^z = 0$ has a solution in $\{z \in \mathbb{C} : |z| \leq 2\}$ Show $z^4 + e^z = 0$ has a solution in $\{z \in \mathbb{C} : |z|  \leq 2\}$.
I would like if in the proof the tools of algebraic topology were preferred over the other tools of analysis, complex analysis, algebra etc.

The tools are the same. Let $f(z)=z^4+e^z$ and let $g(z)=z^4$. Show that $\dfrac{f}{|f|}$ and $\dfrac{g}{|g|}$ are homotopic as maps $\{z: |z|=2\} \to S^1$. Note that if $h\colon S^1\to S^1$ has nonzero degree, then $h$ has a root in $D^2$. (Why?)
