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Here we calculate regular expression of the given machine:

This machine has two final state ${\mathbf{q}}_{\mathbf{1}}$ and ${\mathbf{q}}_{\mathbf{2}}$.

Here for ${\mathbf{q}}_{\mathbf{1}}$ final state we have two distinct path to calculate regular expression so, it sum of two regular expression for two distinct path and same for ${\mathbf{q}}_{\mathbf{2}}$ also.

Regular expression for ${\mathbf{q}}_{\mathbf{1}}$:

RE1 = a(a + aa*b)* + b(a + ba*a)*ba* 

Regular expression for ${\mathbf{q}}_{\mathbf{2}}$:

RE2 = b(a + ba*a)* + a(a + aa*b)*aa* 

Regular expression for machine:

RE = RE1 + RE2

RE = a(a + aa*b)* + b(a + ba*a)*ba* + b(a + ba*a)* + a(a + aa*b)*aa*

= a(a + aa*b)* (λ + aa*) + b(a + ba*a)* (λ + ba*)

= a(a + aa*b)* + b(a + ba*a)* (λ + ba*)         [Therefore, (λ + aa*) = a*]