Simplify. \[ \frac{u x-2 u-x+2}{u^{2}-1} \]

To simplify the expression \(\frac{u x - 2u - x + 2}{u^{2} - 1}\), we will start by factoring both the numerator and the denominator. First, let's factor the denominator \(u^2 - 1\): \[ u^2 - 1 = (u - 1)(u + 1) \] Next, let's focus on the numerator \(ux - 2u - x + 2\). We can rearrange and group the terms: \[ ux - x - 2u + 2 = x(u - 1) - 2(u - 1) \] Now, we can factor out \((u - 1)\) from the numerator: \[ x(u - 1) - 2(u - 1) = (u - 1)(x - 2) \] Now we can rewrite the entire expression: \[ \frac{(u - 1)(x - 2)}{(u - 1)(u + 1)} \] Provided that \(u \neq 1\) (to avoid division by zero), we can cancel \((u - 1)\) from the numerator and denominator: \[ \frac{x - 2}{u + 1} \] Thus, the simplified expression is: \[ \boxed{\frac{x - 2}{u + 1}} \]