Yes, $6m(2m+10)$ is divisible by $4$. To see this, note that $6m(2m+10)=12m(m+5)=4(3m)(m+5)$. Clearly, $4(3m)(m+5)$ is divisible by $4$, so we get that $4\times(3m)(m+5)=6m(2m+10)$, so $4$ divides $6m(2m+10)$ by definition.
Work Step by Step
Recall that $a|b$, where $a$ and $b$ are integers such that $a\ne0$, if and only if $b=ka$ for some integer $k$. Here, $k=3m(m+5)$, which is an integer because $m$, $3$, and $5$ are integers.