Using Myhill-Nerode to prove that language is not regular
Consider two words $u_i = b^i aa, u_j = b^j aa \in \{a, b\}^\ast$ with $i &lt; j$. We have that $u_i \not\sim_L u_j$ because if you concatenate the word $v = b^{i+2}$ to both words we obtain
$$u_i \cdot v = b^i aa \cdot b^{i+2} \in L \text{, but } u_j \cdot v = b^j aa \cdot b^{i+2} \notin L.$$
Thus, $v$ separates $u_i$ and $u_j$ for $L$. Since we can find infinitely many $i, j \in \mathbb{N}$ with $i &lt; j$ we have also infinitely many Myhill-Nerode-equivalence classes.