I am trying to find an DFA for the regular language given by the expression $ L\left( aa^{\ast }\left( a+b\right) \right)$ .

First simplifying $ L\left( aa^{\ast }\left( a+b\right) \right)$ we get

$ L\left( aa^{\ast }\left( a+b\right) \right)$ $ = L\left( a\right) L\left( a^{\ast }\right) L\left( a+b\right) $

Then I constructed an NFA for it , which is given below :

But I am not able to simplify the above NFA to a DFA as the state $ q_1$ has two $ \lambda$ transitions and I am not understanding how to deal with them .