To simplify the complex fraction given:
[tex]\[
\frac{n-6}{n^2 + 11n + 24} \times \frac{n+3}{n+1}
\][/tex]
let's start by factoring the denominator of the first fraction, [tex]\( n^2 + 11n + 24 \)[/tex].
### Step 1: Factor the Quadratic Expression
The expression [tex]\( n^2 + 11n + 24 \)[/tex] can be factored as follows:
- Look for two numbers that multiply to 24 (the constant term) and add to 11 (the coefficient of [tex]\( n \)[/tex]).
- These numbers are 3 and 8.
Therefore,
[tex]\[
n^2 + 11n + 24 = (n + 3)(n + 8)
\][/tex]
### Step 2: Substitute Back into the Expression
Substitute the factored form of the quadratic expression into the original complex fraction:
[tex]\[
\frac{n-6}{(n + 3)(n + 8)} \times \frac{n+3}{n+1}
\][/tex]
### Step 3: Simplify the Expression
To multiply these fractions, first multiply the numerators together and the denominators together:
[tex]\[
\frac{(n-6) \cdot (n+3)}{(n + 3)(n + 8) \cdot (n + 1)}
\][/tex]
Notice that [tex]\( (n+3) \)[/tex] appears in both the numerator and the denominator. We can cancel out the common factor [tex]\( (n+3) \)[/tex]:
[tex]\[
\frac{(n-6)}{(n + 8) \cdot (n + 1)}
\][/tex]
Thus the simplified form of the given expression is:
[tex]\[
\frac{n - 6}{(n + 1)(n + 8)}
\][/tex]
### Conclusion
After simplification, the given fraction simplifies to:
[tex]\[
\frac{n - 6}{(n + 1)(n + 8)}
\][/tex]
Therefore, the correct answer among the provided options is:
[tex]\[
\frac{(n-6)}{(n+1)(n+8)}
\][/tex]
