Show that the Language is irregular
Suppose that $L$ is regular. Then so is $L' = L \circ \{ccc\} = \{a^3 b^n c^n \mid n \ge 3\}$. This implies that $L'' = (aaa)^{-1}L' = \{b^n c^n \mid n \ge 3\}$ is also regular and, in turn, that $L''' = L'' \cup \{\varepsilon, bc, bbcc\} = \{b^n c^n \mid n \in \mathbb{N}\}$ is regular. This is a contradiction since $L'''$ is known not to be regular.