Convert DFA to Regular Expression

In this old exam-task I don't understand all the steps to convert the DFA below to a Regular Expression. The $q_2$ state is eliminated first.
The provided solution to eliminate $q_2$ is:

If we first eliminate $q_2$ we obtain an intermediate automata with $3$ states
$(q_0$,$q_1,q_3)$ such that:

• We go from $q_0$ to $q_1$ with RE $a+ba$

• We go from $q_0$ to $q_3$ with RE $bb$

• We go from $q_1$ to $q_1$ with RE $ba$

• We go from $q_1$ to $q_3$ with RE $a+bb$

I don't understand nr2. $q_3$ can also be reached using RE $aa$. Why is this left out?

:

:

The conversion in each step forms REs that describe

• The previous direct edge from one state to another and

• the path(s) that use(s) only the removed state as an intermediate state.

In your example, the path for $aa$ goes through $q_1$, which is not removed in this step. Thus it is not added to the RE.

StackExchange Computer Science Q#28517, answer score: 8