Final answer:
The simplest form of the product is with the numerator x² - 4 and the denominator x - 1. The expression has an excluded value of x = 1.
Step-by-step explanation:
The expression given is (x² - 3x - 10)/(x² - 6x + 5) × (x - 2)/(x - 5). To simplify this expression, we first factor both numerators and denominators where possible.
Factoring the first numerator: x² - 3x - 10 = (x - 5)(x + 2).
Factoring the first denominator: x² - 6x + 5 = (x - 5)(x - 1).
Substituting the factored forms back into the expression, we get: ((x - 5)(x + 2))/((x - 5)(x - 1)) × (x - 2)/(x - 5).
Now we can cancel out the common terms. The (x - 5) terms cancel out in the first fraction and again between the fractions leaving us with: (x + 2)/(x - 1) × (x - 2). Simplifying further, we get (x + 2)(x - 2)/(x - 1), which simplifies to x² - 4 in the numerator and x - 1 in the denominator since (x + 2)(x - 2) is the difference of squares.
The expression has an excluded value where the denominator equals zero. Since our simplified denominator is x - 1, setting this equal to zero gives us x = 1 as the excluded value.