To find the product of the given rational expressions, let's follow a structured step-by-step process.
The rational expressions are:
[tex]\[
\frac{x}{x+3} \quad \text{and} \quad \frac{x}{x+2}
\][/tex]
Step 1: Multiply the numerators together.
The numerators of the given expressions are [tex]\(x\)[/tex] and [tex]\(x\)[/tex]:
[tex]\[
x \cdot x = x^2
\][/tex]
Step 2: Multiply the denominators together.
The denominators of the given expressions are [tex]\(x+3\)[/tex] and [tex]\(x+2\)[/tex]:
[tex]\[
(x+3) \cdot (x+2)
\][/tex]
Step 3: Write the product as a single expression.
Combining the results from steps 1 and 2, we get:
[tex]\[
\frac{x^2}{(x+3)(x+2)}
\][/tex]
Step 4: Simplify the denominator (if possible).
Let's expand the denominator [tex]\( (x+3)(x+2) \)[/tex]:
[tex]\[
(x+3)(x+2) = x(x+2) + 3(x+2) = x^2 + 2x + 3x + 6 = x^2 + 5x + 6
\][/tex]
Step 5: Rewrite the expression using the expanded denominator:
[tex]\[
\frac{x^2}{x^2 + 5x + 6}
\][/tex]
Therefore, the product of the rational expressions [tex]\(\frac{x}{x+3} \cdot \frac{x}{x+2}\)[/tex] is:
[tex]\[
\frac{x^2}{x^2 + 5x + 6}
\][/tex]
Finally, based on the given options, the correct answer is:
[tex]\[
\boxed{\frac{x^2}{x^2+5 x+6}}
\][/tex]
So the correct answer is [tex]\( \text{Option A} \)[/tex].
