\(\frac{a.\left(b+c\right)-b-1}{b.\left(a-1\right)+a-1}=\frac{ab+ac-b-1}{ab-b+a-1}\)
\(=\frac{ab-b-1+a+\left(a-1\right).c}{ab-b+a-1}=1+\frac{\left(a-1\right).c}{ab-b+a-1}\)
\(=1+\frac{\left(a-1\right).c}{b.\left(a-1\right)+\left(a-1\right)}=1+\frac{\left(a-1\right).c}{\left(b+1\right).\left(a-1\right)}\)