Regular Expression with at least one a, even number of b
Another way of seeing this which gives a factorized representation is to try to locate the first $a$. It appears after either an odd or an even number of $b$. 

If it appears after an even number of $b$ then you only have to check that what follows also has an even number of $b$ which can be done with Yuval's expression $a^*(ba^*ba^*)^*$. 

If the first $a$ comes after an odd number of $b$, then you also have to find another $b$ and then check that the rest contains an even number of $b$.

This could be translated as follows: $(bb)^*(a+ba^+b)a^*(ba^*ba^*)^*$. Again, you can replace $a^+$ by $aa^*$.