Prove that the language L = {a^(m+n) b^m a^n | m, n ≥ 0} ∪ {a^m b^n a^(m+n) | m, n ≥ 0} is not regular
Regular languages are closed under finite intersection, use that to your advantage. In this case, the reason the language isn't regular is because it has a $K = \{a^n b^n | n \geq 0 \}$ hidden inside it, pull it out using a regular language as a filter. More explicitly, notice that $K = L \cap a^*b^*$.