To simplify the given rational expression
[tex]\[
\frac{5(x-2)}{(x-2)(x+4)},
\][/tex]
we need to factor and then cancel out any common terms in the numerator and the denominator.
Step-by-step process:
1. Identify common factors: Notice that the numerator [tex]\(5(x-2)\)[/tex] and the denominator [tex]\((x-2)(x+4)\)[/tex] both have the factor [tex]\((x-2)\)[/tex].
2. Cancel the common factor: Since [tex]\(x \neq 2\)[/tex] and [tex]\(x \neq -4\)[/tex], we can safely cancel [tex]\((x-2)\)[/tex] from both the numerator and the denominator.
[tex]\[
\frac{5(x-2)}{(x-2)(x+4)} = \frac{5 \cancel{(x-2)}}{\cancel{(x-2)} (x+4)}
\][/tex]
This leaves us with:
[tex]\[
\frac{5}{x+4}
\][/tex]
So, the simplified expression is:
[tex]\[
\frac{5}{x+4}
\][/tex]
Among the given options, this corresponds to:
D. [tex]\(\frac{5}{x+4}\)[/tex]
Therefore, the correct answer is [tex]\(\boxed{D}\)[/tex].
