Question: User: $\frac{x^2 - 1}{x(x + 2)} \times y \times \frac{3x}{x^2 + 2x}$
1. **State the problem:** Simplify the expression $$\frac{x^2 - 1}{x(x + 2)} \times y \times \frac{3x}{x^2 + 2x}$$.
2. **Recall important formulas and rules:**
- Factor differences of squares: $$x^2 - 1 = (x - 1)(x + 1)$$.
- Factor common terms in denominators: $$x^2 + 2x = x(x + 2)$$.
- When multiplying fractions, multiply numerators together and denominators together.
- Cancel common factors between numerator and denominator.
3. **Rewrite the expression with factored terms:**
$$\frac{(x - 1)(x + 1)}{x(x + 2)} \times y \times \frac{3x}{x(x + 2)}$$
4. **Combine the fractions:**
$$\frac{(x - 1)(x + 1)}{x(x + 2)} \times \frac{3x}{x(x + 2)} \times y = \frac{(x - 1)(x + 1) \times 3x}{x(x + 2) \times x(x + 2)} \times y$$
5. **Simplify the denominator:**
$$x(x + 2) \times x(x + 2) = x^2 (x + 2)^2$$
6. **Write the full expression:**
$$\frac{3x (x - 1)(x + 1)}{x^2 (x + 2)^2} \times y$$
7. **Cancel common factors:**
Cancel one $x$ from numerator and denominator:
$$\frac{3\cancel{x} (x - 1)(x + 1)}{\cancel{x} x (x + 2)^2} \times y = \frac{3 (x - 1)(x + 1)}{x (x + 2)^2} \times y$$
8. **Final simplified expression:**
$$\frac{3 (x - 1)(x + 1)}{x (x + 2)^2} y$$
This is the simplified form of the original expression.