No it cannot be expressed in terms of regular expressions. That is because, your expression, requires a number of "aa" followed by equal number of "bc". This requires infinite memory. FA do not have infinite memory.
It can be expressed in context-free grammar.-
S -> aaSbc | έ Epsilon stands for string of zero length(empty string).
This generates strings like - Valid string - έ(Empty string), aabc, aaaabcbc and so on.
Read more about context free and regular grammar here.