write the state equations
$A= \epsilon + Ba$
$B= A\epsilon + C(a+b)$
$C= Aa + Cb$
Apply Arden's Theorem $[\text{if}\; R= Q+RP\; \text{then} \;R = QP*]$
$C= Aab^*$
put equation of $C$ in $B$
$B= A(\epsilon +ab^*(a+b))$
put equation of $B$ in $A$
$A=\epsilon+ A(\epsilon +ab^*(a+b))a$
Apply Arden's Theorem
$A=((\epsilon +ab^*(a+b))a)^*$
put equation of $A$ in $B$
so $B=((\epsilon +ab^*(a+b))a)^*(\epsilon+ab^*(a+b))$