To determine which of the provided options represents the product of the rational expressions [tex]\(\frac{x+2}{x-4} \cdot \frac{3 x}{x+4}\)[/tex], let us break down and simplify the given expressions step-by-step.
1. Identify and Multiply the Numerators:
The numerators of the given fractions are [tex]\(x + 2\)[/tex] and [tex]\(3x\)[/tex].
[tex]\[
(x + 2) \cdot (3x)
\][/tex]
When we multiply these numerators, we distribute the terms inside the parentheses:
[tex]\[
x \cdot 3x + 2 \cdot 3x = 3x^2 + 6x
\][/tex]
2. Identify and Multiply the Denominators:
The denominators of the given fractions are [tex]\(x - 4\)[/tex] and [tex]\(x + 4\)[/tex].
[tex]\[
(x - 4) \cdot (x + 4)
\][/tex]
Notice that this is a difference of squares:
[tex]\[
(x - 4)(x + 4) = x^2 - 4^2 = x^2 - 16
\][/tex]
3. Construct the Combined Rational Expression:
Using the results from steps 1 and 2, the combined rational expression will be:
[tex]\[
\frac{3x^2 + 6x}{x^2 - 16}
\][/tex]
Now, let’s match our simplified expression against the provided answer choices:
A. [tex]\(\frac{3 x^2+6 x}{x^2-16}\)[/tex]
B. [tex]\(\frac{3 x^2+2 x}{x-16}\)[/tex]
C. [tex]\(\frac{3 x+6}{x^2-16}\)[/tex]
D. [tex]\(\frac{4 x+2}{x^2}\)[/tex]
Clearly, option A matches the simplified form perfectly.
Thus, the product of the given rational expressions is:
[tex]\[
\boxed{\frac{3 x^2+6 x}{x^2-16}}
\][/tex]
So, the correct choice is A.
