Add. \[ \frac{8}{x-4}+\frac{x-5}{x-1} \] Simplify your answer as much as possible.

To add the fractions \(\frac{8}{x-4} + \frac{x-5}{x-1}\), we first need a common denominator, which is \((x-4)(x-1)\). We will rewrite both fractions with this common denominator. Rewriting the first fraction: \[ \frac{8}{x-4} = \frac{8(x-1)}{(x-4)(x-1)} = \frac{8x - 8}{(x-4)(x-1)} \] Rewriting the second fraction: \[ \frac{x-5}{x-1} = \frac{(x-5)(x-4)}{(x-1)(x-4)} = \frac{(x^2 - 4x - 5x + 20)}{(x-4)(x-1)} = \frac{x^2 - 9x + 20}{(x-4)(x-1)} \] Now we can combine the two fractions: \[ \frac{8x - 8 + x^2 - 9x + 20}{(x-4)(x-1)} \] Combine like terms in the numerator: \[ 8x - 9x + x^2 - 8 + 20 = x^2 - x + 12 \] So we have: \[ \frac{x^2 - x + 12}{(x-4)(x-1)} \] This is the simplified expression: \[ \frac{x^2 - x + 12}{(x-4)(x-1)} \]