Let's find the product of the given rational expressions step-by-step:
We need to multiply:
[tex]\[
\frac{3}{x+2} \cdot \frac{7}{2x}
\][/tex]
### Step 1: Multiply the numerators
First, multiply the numerators of the two fractions:
[tex]\[
3 \times 7 = 21
\][/tex]
### Step 2: Multiply the denominators
Next, multiply the denominators of the two fractions:
[tex]\[
(x+2) \times (2x)
\][/tex]
To expand this, we distribute [tex]\(2x\)[/tex] across [tex]\(x + 2\)[/tex]:
[tex]\[
2x \times x + 2x \times 2 = 2x^2 + 4x
\][/tex]
### Step 3: Form the new fraction
Now, combine the results from the numerator and the denominator:
[tex]\[
\frac{21}{2x^2 + 4x}
\][/tex]
### Step 4: Simplify the fraction (if possible)
In this case, the numerator [tex]\(21\)[/tex] and the denominator [tex]\(2x^2 + 4x\)[/tex] do not have any common factors other than 1. Therefore, the fraction cannot be simplified further.
### Conclusion
Therefore, the product of the rational expressions [tex]\(\frac{3}{x+2}\)[/tex] and [tex]\(\frac{7}{2x}\)[/tex] is:
[tex]\[
\frac{21}{2x^2 + 4x}
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{\frac{21}{2x^2 + 4x}} \][/tex]
Which matches:
A. [tex]\(\frac{21}{2 x^2+4 x}\)[/tex]
