simplify regex, [star] mysteriously disappears

Question

! Problem This is my lecture notes;"Prove a*(b+ab*)=b+aab"
I am having trouble understanding what happened at line 3-> line 4;
What I understand Two things happen at those two line;

1. Take ab* out of term ab* and term aa*b, results(Λ+aa*);
2. Last term aa*ab* simplifies to aa*b* Because a in the middle is redundant;

Question is what happen to the * at aa*b(*)?

Answer 1

It looks like your professor left out lines 3⅓ and 3⅔:

Line 3:     b + ab* + aa*b + aa*ab*
Line 3⅓:    b + ab* + aa*ab* + aa*b      ←  commute last two terms
Line 3⅔:    b + Λab* + aa*ab* + aa*b     ←  ab* = Λab*
Line 4:     b + (Λ + aa*)ab* + aa*b      ←  distributive property