To divide the given rational expressions and simplify the quotient, follow these steps:
1. Write the division as multiplication:
[tex]\[
\frac{x^2 - 4x - 45}{x+6} \div \frac{x^2 - 3x - 40}{x+6} = \frac{x^2 - 4x - 45}{x+6} \times \frac{x+6}{x^2 - 3x - 40}
\][/tex]
2. Simplify the expression:
Notice that multiplying by [tex]\(\frac{x+6}{x+6}\)[/tex] is essentially multiplying by 1, so we can simplify:
[tex]\[
\frac{x^2 - 4x - 45}{x+6} \times \frac{x+6}{x^2 - 3x - 40} = \frac{x^2 - 4x - 45}{x^2 - 3x - 40}
\][/tex]
3. Factorize the numerators and denominators:
- Factorize [tex]\(x^2 - 4x - 45\)[/tex]:
[tex]\[
x^2 - 4x - 45 = (x - 9)(x + 5)
\][/tex]
- Factorize [tex]\(x^2 - 3x - 40\)[/tex]:
[tex]\[
x^2 - 3x - 40 = (x - 8)(x + 5)
\][/tex]
4. Substitute the factored forms back into the expression:
[tex]\[
\frac{(x - 9)(x + 5)}{(x - 8)(x + 5)}
\][/tex]
5. Cancel out the common factors:
Since [tex]\( (x + 5) \)[/tex] appears in both the numerator and denominator, we can cancel them out:
[tex]\[
\frac{x - 9}{x - 8}
\][/tex]
Therefore, the simplified quotient of the given expression is:
[tex]\[
\boxed{\frac{x - 9}{x - 8}}
\][/tex]
