Which of the following is the product of the rational expressions shown below?

[tex]\[ \frac{x}{x+3} \cdot \frac{x}{x+2} \][/tex]

A. [tex]\[ \frac{x^2}{x^2+5x+6} \][/tex]

B. [tex]\[ \frac{x^2}{x^2+6} \][/tex]

C. [tex]\[ \frac{2x}{x^2+5x+6} \][/tex]

D. [tex]\[ \frac{x^2}{x^2+5} \][/tex]



Answer :

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].