\( \left. \begin{array} { l | l } { } & { ( x - 1 ) ^ { - 1 } \times ( 1 - x ^ { 2 } ) } \\ { = } & { \frac { ( 1 - x ) ( 1 + x ) } { x - 1 } } \\ { = - ( 1 + x ) } \end{array} \right. \)

Did you know that the expression you’ve worked on showcases a fascinating relationship between algebraic manipulation and the concept of inverses? By factoring and simplifying, you not only simplify the expression but also reveal how cancellation can transform one term into another, making your calculations cleaner. This is crucial in many areas of mathematics, particularly in calculus where handling limits is often done by simplifying complex expressions. When simplifying fractions like the one you've illustrated, watch out for signs! It's easy to overlook negative signs, especially when juggling multiple factors. Remember, \( -(1 + x) \) means you’re taking the whole expression and flipping its sign, which can sometimes lead to mistakes if you forget to distribute the negative. Double-check each step to ensure you don’t inadvertently end up with the wrong sign in your final answer!