Sure, let's solve the given expression step-by-step.
We need to simplify the expression [tex]\(\frac{n^2 - 7n + 12}{(n - 3)(n + 4)}\)[/tex] and determine which of the given options it matches. Here’s the detailed solution:
1. Factor the Numerator:
We start with the numerator [tex]\(n^2 - 7n + 12\)[/tex]. We need to factorize it.
[tex]\[
n^2 - 7n + 12
\][/tex]
To factorize this quadratic expression, we look for two numbers that multiply to [tex]\(12\)[/tex] (the constant term) and add to [tex]\(-7\)[/tex] (the coefficient of [tex]\(n\)[/tex]).
These numbers are [tex]\(-3\)[/tex] and [tex]\(-4\)[/tex], since:
[tex]\[
-3 \times -4 = 12
\][/tex]
[tex]\[
-3 + (-4) = -7
\][/tex]
Therefore, we can write the numerator as:
[tex]\[
n^2 - 7n + 12 = (n - 3)(n - 4)
\][/tex]
2. Simplify the Expression:
Now, substitute the factorized form of the numerator back into the original expression:
[tex]\[
\frac{(n - 3)(n - 4)}{(n - 3)(n + 4)}
\][/tex]
Notice that both the numerator and the denominator have a common factor, [tex]\((n - 3)\)[/tex]. As long as [tex]\(n \neq 3\)[/tex], we can cancel this common factor:
[tex]\[
\frac{(n - 3)(n - 4)}{(n - 3)(n + 4)} = \frac{n - 4}{n + 4}
\][/tex]
3. Result:
After cancelling the common factor, we have:
[tex]\[
\frac{n - 4}{n + 4}
\][/tex]
Hence, the simplified form of the given expression [tex]\(\frac{n^2 - 7n + 12}{(n - 3)(n + 4)}\)[/tex] is [tex]\(\frac{n - 4}{n + 4}\)[/tex].
Therefore, the correct option that matches this simplified expression is:
[tex]\[
\boxed{\frac{n - 4}{n + 4}}
\][/tex]
Which corresponds to the option:
D. [tex]\(\frac{n - 4}{n + 4}\)[/tex]
