To find the product of the given rational expressions, we will follow the steps of multiplication and simplification:
Let's start with the given rational expressions:
[tex]\[ \frac{x}{x-3} \cdot \frac{2x}{x+5} \][/tex]
### Step 1: Multiply the numerators and the denominators
Multiply the numerators together:
[tex]\[ x \cdot 2x = 2x^2 \][/tex]
Multiply the denominators together:
[tex]\[ (x-3)(x+5) \][/tex]
So, the new rational expression becomes:
[tex]\[ \frac{2x^2}{(x-3)(x+5)} \][/tex]
### Step 2: Simplify the denominator
To simplify the denominator, we use the distributive property (FOIL method):
[tex]\[ (x-3)(x+5) = x \cdot x + x \cdot 5 - 3 \cdot x - 3 \cdot 5 \][/tex]
[tex]\[ = x^2 + 5x - 3x - 15 \][/tex]
[tex]\[ = x^2 + 2x - 15 \][/tex]
Now we substitute the simplified form of the denominator back into our expression:
[tex]\[ \frac{2x^2}{x^2 + 2x - 15} \][/tex]
### Step 3: Match the result with the given options
We can see that the simplified form of the product is:
[tex]\[ \frac{2x^2}{x^2 + 2x - 15} \][/tex]
Comparing this with the provided answer choices:
A. [tex]\(\frac{x^2}{x^2-2x+15}\)[/tex]
B. [tex]\(\frac{2x^2}{x^2+2}\)[/tex]
C. [tex]\(\frac{2x^2}{x^2+2x-15}\)[/tex]
D. [tex]\(\frac{3x}{2x+2}\)[/tex]
Our simplified expression matches option C.
### Conclusion
The correct answer is:
[tex]\[ \boxed{\frac{2x^2}{x^2 + 2x - 15}} \][/tex]
