Using closure properties, prove that $L=\{a^kb^ra^m|k,r,m\ge0 \text{ and } m=k+r\}$ is not regular
Consider the intersection of your language $L$ with the regular language $L' = \{b^r a^m \mid r,m \ge 0 \}$.
Then $L \cap L'$ contains all the words of the form $a^k b^r a^m$ where $k=0$ and $m=k+r=r$. That is, $L \cap L' = \{ b^r a^r \mid r \ge 0  \}$, which is a well-known non-regular language.