Verified answer

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    (x³-x²+x-1)/(x²-1)

         x²(x-1)+ x - 1               ← factored out x² to get the (x-1)'s

= ———————

                   (x²-1)

         x²(x-1)+ (x-1)

= ———————

                   (x²-1)                             ← notice the difference of squares

         (x²+1)(x-1)                       ← factored out (x-1)

= ——————

           (x+1)(x-1)                       ← factored the difference of squares

         (x²+1)(x∕-∕1)

= ——————                       ← now, cancel the (x-1)'s

           (x+1)(x∕-∕1)

          (x²+1)

= ————                                  ← ANSWER

             x+1

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x² - 1 = (x+1)(x-1).

Using synthetic division to divide 1x³ - 1x² + 1x - 1 by (x-1):

1__-1__1__-1__| 1

___1__0__1

1__ 0__1__0

shows that x³ - x² + x - 1 = (x-1)(x² + 1).

Therefore the expression reduces to (x²+1)/(x+1).