Sure, let's solve the problem step-by-step to find the simplest form of the given product and determine any excluded values.
We begin with the given product:
[tex]\[
\frac{x^2 - 3x - 10}{x^2 - 6x + 5} \cdot \frac{x - 2}{x - 5}
\][/tex]
First, we factorize the expressions in both the numerator and the denominator.
### Step 1: Factorize the numerator and the denominator
Factorize [tex]\(x^2 - 3x - 10\)[/tex]:
[tex]\[
x^2 - 3x - 10 = (x - 5)(x + 2)
\][/tex]
Factorize [tex]\(x^2 - 6x + 5\)[/tex]:
[tex]\[
x^2 - 6x + 5 = (x - 1)(x - 5)
\][/tex]
Now, substitute the factored forms into the product:
[tex]\[
\frac{(x - 5)(x + 2)}{(x - 1)(x - 5)} \cdot \frac{x - 2}{x - 5}
\][/tex]
### Step 2: Simplify the expression
Combine the numerator and the denominator:
[tex]\[
\frac{(x - 5)(x + 2)(x - 2)}{(x - 1)(x - 5)(x - 5)}
\][/tex]
Notice that the factor [tex]\((x - 5)\)[/tex] appears in both the numerator and the denominator. We can cancel one [tex]\((x - 5)\)[/tex]:
[tex]\[
\frac{(x + 2)(x - 2)}{(x - 1)(x - 5)}
\][/tex]
The numerator [tex]\((x + 2)(x - 2)\)[/tex] simplifies to:
[tex]\[
x^2 - 4
\][/tex]
Thus, the simplified form is:
[tex]\[
\frac{x^2 - 4}{(x - 1)(x - 5)}
\][/tex]
### Step 3: Identify excluded values
The expression is undefined where the denominator is zero. We have:
[tex]\[
(x - 1)(x - 5) = 0
\][/tex]
Solving for [tex]\(x\)[/tex], we get:
[tex]\[
x = 1 \quad \text{or} \quad x = 5
\][/tex]
### Final answer
Therefore, the simplest form of the given product is:
[tex]\[
\frac{x^2 - 4}{(x - 1)(x - 5)}
\][/tex]
The expression has excluded values at:
[tex]\[
x = 1 \quad \text{or} \quad x = 5
\][/tex]
So, the correct answers for the given drop-down menus are:
The simplest form of this product has a numerator of [tex]\(x^2 - 4\)[/tex] and a denominator of [tex]\((x - 1)(x - 5)\)[/tex]. The expression has an excluded value of [tex]\(x = 5 \vee x = 1\)[/tex].
