Sure, let's work through the problem step-by-step to find the simplified product of the given fractions.
### Given Expression
First, let's write down the given fractions:
1. [tex]\(\frac{25 x^2}{2(x + 3)}\)[/tex]
2. [tex]\(\frac{2}{5x}\)[/tex]
### Step-by-Step Solution
#### Step 1: Write the product of the fractions
We write the product of the two fractions by multiplying their numerators together and their denominators together:
[tex]\[
\frac{25 x^2}{2(x + 3)} \cdot \frac{2}{5 x}
\][/tex]
#### Step 2: Multiply the numerators
Multiply the numerators of both fractions:
[tex]\[
25 x^2 \cdot 2 = 50 x^2
\][/tex]
#### Step 3: Multiply the denominators
Multiply the denominators of both fractions:
[tex]\[
2(x + 3) \cdot 5x = 10x(x + 3)
\][/tex]
#### Step 4: Combine the products
Now, our fraction becomes:
[tex]\[
\frac{50 x^2}{10 x (x + 3)}
\][/tex]
#### Step 5: Simplify the fraction
We can simplify the fraction by canceling out common factors in the numerator and denominator. Notice that [tex]\(50 x^2\)[/tex] and [tex]\(10 x\)[/tex] share a common factor of [tex]\(10 x\)[/tex]:
[tex]\[
\frac{50 x^2}{10 x (x + 3)} = \frac{50 x^2}{10 x (x + 3)} = \frac{5x}{x + 3}
\][/tex]
So, the simplified product is:
[tex]\[
\frac{5 x}{x + 3}
\][/tex]
### Final Answer
From the given choices:
- [tex]\(\frac{1}{x+3}\)[/tex]
- [tex]\(\frac{2}{x+3}\)[/tex]
- [tex]\(\frac{5 x}{x+3}\)[/tex]
- [tex]\(\frac{10 x}{x+3}\)[/tex]
The correct simplified product is:
[tex]\[
\boxed{\frac{5 x}{x + 3}}
\][/tex]
