To find the product of the rational expressions [tex]\(\frac{x+2}{x-4} \cdot \frac{3 x}{x+4}\)[/tex], we follow these steps:
1. Multiply the numerators together:
[tex]\[
(x + 2) \cdot (3x)
\][/tex]
2. Multiply the denominators together:
[tex]\[
(x - 4) \cdot (x + 4)
\][/tex]
Let's start with the numerator:
[tex]\[
(x + 2)(3x) = 3x(x + 2) = 3x^2 + 6x
\][/tex]
Now for the denominator, notice that we have a difference of squares:
[tex]\[
(x - 4)(x + 4) = x^2 - 16
\][/tex]
Putting these together, we get the product of the rational expressions:
[tex]\[
\frac{(x + 2) \cdot (3x)}{(x - 4) \cdot (x + 4)} = \frac{3x^2 + 6x}{x^2 - 16}
\][/tex]
Hence, the correct option is:
[tex]\[
\mathbf{A.} \frac{3x^2 + 6x}{x^2 - 16}
\][/tex]
