Certainly, let's find the product of the given rational expressions step by step.
The problem provides two rational expressions:
[tex]\[
\frac{3x}{x+1} \quad \text{and} \quad \frac{x}{x-7}
\][/tex]
To multiply these rational expressions, we follow these steps:
1. Multiply the numerators: Multiply the numerators of both expressions together.
2. Multiply the denominators: Multiply the denominators of both expressions together.
3. Simplify the resulting expression.
First, let's multiply the numerators:
[tex]\[
3x \cdot x = 3x^2
\][/tex]
Next, let's multiply the denominators:
[tex]\[
(x+1) \cdot (x-7) = (x+1)(x-7)
\][/tex]
So the product of the rational expressions before simplification is:
[tex]\[
\frac{3x^2}{(x+1)(x-7)}
\][/tex]
Now we need to simplify the denominator. Let's expand [tex]\( (x+1)(x-7) \)[/tex]:
[tex]\[
(x+1)(x-7) = x \cdot x + x \cdot (-7) + 1 \cdot x + 1 \cdot (-7) \\
= x^2 - 7x + x - 7 \\
= x^2 - 6x - 7
\][/tex]
Now we can rewrite the rational expression with the expanded denominator:
[tex]\[
\frac{3x^2}{x^2 - 6x - 7}
\][/tex]
After simplification, the final product of the rational expressions is:
[tex]\[
\frac{3x^2}{x^2-6x-7}
\][/tex]
This matches with option A. Therefore, the correct answer is:
[tex]\[
\boxed{\frac{3 x^2}{x^2-6 x-7}}
\][/tex]
