(a* + b)* is equivalent to:
- $\text{(a + b)*}$
- $\text{(a + b*)*}$
- $\text{(a*b*)*}$
- basically any regular expression which can generate all strings in $\Sigma^*$.
Now looking at the options:
(a) $\text{a*b*}$ - This can't generate abab
(b) $\text{(a*b+b)*}$ - This can't generate aaa (at least one b needs to be there in any non-empty string)
(c) $\text{(a+b*)*}$ - This one can generate all strings, i.e it is equivalent to (a+b)*
(d) $\text{(a*b)*}$ - This can't generate aaa (at least one b needs to be there in any non-empty string)
So, correct option should be (C) (a+b*)*.