To find the product of the given rational expressions:
[tex]\[ \frac{x+5}{x+7} \cdot \frac{10x}{x-7} \][/tex]
we will multiply the numerators together and the denominators together.
### Step-by-Step Solution:
1. Write down the given expressions:
[tex]\[ \frac{x+5}{x+7} \quad \text{and} \quad \frac{10x}{x-7} \][/tex]
2. Multiply the numerators:
[tex]\[ (x + 5) \cdot 10x = 10x(x + 5) \][/tex]
3. Multiply the denominators:
[tex]\[ (x + 7) \cdot (x - 7) \][/tex]
4. Combine the results to form one fraction:
[tex]\[ \frac{10x(x + 5)}{(x + 7)(x - 7)} \][/tex]
5. Simplify the denominator using the difference of squares formula:
[tex]\[ (x + 7)(x - 7) = x^2 - 49 \][/tex]
6. Re-write the product with the simplified denominator:
[tex]\[ \frac{10x(x + 5)}{x^2 - 49} \][/tex]
7. Expanding the numerator (optional for clarity):
[tex]\[ 10x(x + 5) = 10x^2 + 50x \][/tex]
8. The final simplified product of the rational expressions is then:
[tex]\[ \frac{10x^2 + 50x}{x^2 - 49} \][/tex]
### Conclusion:
Upon reviewing the options provided:
A. [tex]\(\frac{10x^2 + 50x}{x^2 - 49}\)[/tex]
B. [tex]\(\frac{10x^2 + 50x}{x^2 - 14}\)[/tex]
C. [tex]\(\frac{10x^2 + 5}{x^2 - 49}\)[/tex]
D. [tex]\(\frac{10x^2 + 50x}{x^2}\)[/tex]
The correct answer is:
[tex]\[ \boxed{\frac{10x^2 + 50x}{x^2 - 49}} \][/tex]
which matches option A.
