Question

which rational expression represents \\(\frac{x^2 - 9}{3x^2} cdot \frac{3x^3 + 9x^2}{x - 3}\\) when simplified if \\(x \
eq 0\\) or \\(3\\)?

\\(\circ\\) \\((x - 3)(x + 3)\\)

\\(\circ\\) \\(\frac{x - 3}{x + 3}\\)

\\(\circ\\) \\((x + 3)^2\\)

\\(\circ\\) \\(\frac{x + 3}{x - 3}\\)

Explanation:

Step1: Factor numerators/denominators

Factor $x^2-9$ as $(x-3)(x+3)$, and factor $3x^3+9x^2$ as $3x^2(x+3)$.
The expression becomes:
$\frac{(x-3)(x+3)}{3x^2} \cdot \frac{3x^2(x+3)}{x-3}$

Step2: Cancel common factors

Cancel $3x^2$, $(x-3)$ from numerator and denominator:
$\frac{\cancel{(x-3)}(x+3)}{\cancel{3x^2}} \cdot \frac{\cancel{3x^2}(x+3)}{\cancel{x-3}} = (x+3)(x+3)$

Step3: Simplify the product

Multiply the remaining terms:
$(x+3)(x+3) = (x+3)^2$

Answer:

C. $(x + 3)^2$