Rouché's Theorem Let $f,g: U \to \mathbb{C} $ be analytic on $U \subset \mathbb{C} $ open, and let $\gamma $ be any closed curve such that $\gamma \subset U$. If $\vert f(z) \vert \leq \vert g(z) \vert $ on $\gamma $ then $f(z)$ and $f(z) = g(z)$ have the same number of zeros inside $\gamma $.
Let $g(z) = z^6 + 10$ and $h(z) = -5z^4$. Note that $f(z) = g(z) + h(z)$ and \begin{align*} \left\vert h(z) \right\vert = \left\vert -5z^4 \right\vert = 5\vert z \vert ^4 = 80 \end{align*} on $|z|=2$. On the other hand, \begin{align*} \vert g(z) \vert = \left\vert z^6 + 10 \right\vert \leq \vert z \vert ^6 + 10 \leq 42. \end{align*} Therefore, $\vert g(z) \vert \leq \vert h(z) \vert $ on $\vert z \vert = 2$, so by Rouché's Theorem, the number of zeros of $h(z)$ and $g(z) + h(z)$ will be same inside $|z|=2$. Since $h(z)$ has $4$ zeros, $f(z) = g(z) = h(z)$ will also have $4$ roots in $\vert z \vert =2$.