Let's carefully simplify the given quotient step by step.
Given quotient:
[tex]\[
\frac{x^2-2x-8}{x^2-2x-15} \div \frac{2x^2-8x}{2x^2-10x}
\][/tex]
### Step 1: Write the division as a multiplication by the reciprocal
To divide by a fraction, we multiply by its reciprocal:
[tex]\[
\frac{x^2-2x-8}{x^2-2x-15} \times \frac{2x^2-10x}{2x^2-8x}
\][/tex]
### Step 2: Factor each expression in the numerator and denominator
First, factorize each quadratic expression where possible:
Numerators:
1. [tex]\( x^2 - 2x - 8 \)[/tex]:
[tex]\[
x^2 - 2x - 8 = (x - 4)(x + 2)
\][/tex]
2. [tex]\( 2x^2 - 10x \)[/tex]:
[tex]\[
2x^2 - 10x = 2x(x - 5)
\][/tex]
Denominators:
1. [tex]\( x^2 - 2x - 15 \)[/tex]:
[tex]\[
x^2 - 2x - 15 = (x - 5)(x + 3)
\][/tex]
2. [tex]\( 2x^2 - 8x \)[/tex]:
[tex]\[
2x^2 - 8x = 2x(x - 4)
\][/tex]
### Step 3: Substitute the factored forms back into the quotient
[tex]\[
\frac{(x - 4)(x + 2)}{(x - 5)(x + 3)} \times \frac{2x(x - 5)}{2x(x - 4)}
\][/tex]
### Step 4: Simplify the expression by canceling common factors
Observe that several terms cancel out:
- [tex]\((x - 4)\)[/tex] in numerator and denominator
- [tex]\(2x\)[/tex] in numerator and denominator
- [tex]\((x - 5)\)[/tex] in numerator and denominator
After cancelation, we have:
[tex]\[
\frac{(x + 2)}{(x + 3)}
\][/tex]
### Step 5: Determine the simplified numerator and denominator
The simplified form of the quotient is:
[tex]\[
\frac{x + 2}{x + 3}
\][/tex]
### Step 6: Identify values for which the expression does not exist
The expression does not exist when the denominator equals zero. Thus:
[tex]\[
x + 3 = 0 \implies x = -3
\][/tex]
### Final Answer
- The simplified numerator is [tex]\( x + 2 \)[/tex].
- The expression does not exist when [tex]\( x = -3 \)[/tex].
- The simplified denominator is [tex]\( x + 3 \)[/tex].
