I'm trying to write a regular expression over the alphabet $\{a, b\}$ for the language in which every $b$ is preceded and followed by an even number of $a$'s.
I think the regular expression should include something like $((aa)^*b(aa)^*)^*$ but I don't know how to extend it in order to get to the correct form. Can anyone help me by providing a step-by-step solution?
The answer you already have covers almost all the cases: any time there's a $b$, there need to be an even number of $a$'s, so that covers all words containing a $b$.
However, there's an edge case where there are no $b$'s in your word, so you need to union your language with the case where the word has no $b$'s.
$((aa)^*b(aa)^*)^* \cup a^*$ should fit your final answer.
