I4.1 Suppose $a+\frac{1}{a+1}=b+\frac{1}{b-1}-2$, where $a \neq-1, b \neq 1$, and $a-b+2 \neq 0$. Given that $a b-a+b=P$, find the value of $P$.
$\begin{array}{l}a-b+2+\frac{1}{a+1}-\frac{1}{b-1}=0 \\ (a-b+2)\left[1-\frac{1}{(a+1)(b-1)}\right]=0 \\ \Rightarrow a b+b-a-2=0 \\ P=2\end{array}$