A little help with regular expressions
Some things I learnt while learning to answer this type of question:


Negation isn't easy to express. There are no shortcuts. The resulting expressions may be far more complex than the expression you would get if negation was a basic operator in the expression language - in which case you could write $\neg(.^*bbc.^*)$, where $.$ is a shortcut for $(a\cup\neg a)$
Lacking $\neg$, the only way to say $\neg a$ for some symbol $a$ is to enumerate all other symbols: $\neg a = (b\cup c \cup d \cup \ldots)$. In Unix-like regular expressions, you can of course use [^a].
In compound expressions, you can use left factorization.
E.g. $\neg(bbc) = \epsilon \cup \neg b\neg(bc) = \epsilon \cup \neg b(\epsilon \cup \neg c) = \ldots$.
The key insight is that this can be made to work with Kleene star expressions, too; I suggest you invent or look up the details yourself.