1. **Problem statement:** Simplify the expression $$\frac{(x+9)^2}{x} \cdot \frac{x^2}{2x + 18}$$ 2. **Formula and rules:** When multiplying fractions, multiply numerators together and denominators together: $$\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}$$ 3. **Step 1: Factor expressions where possible.** - Note that $$2x + 18 = 2(x + 9)$$ 4. **Step 2: Write the expression with factored terms:** $$\frac{(x+9)^2}{x} \cdot \frac{x^2}{2(x+9)}$$ 5. **Step 3: Multiply numerators and denominators:** $$\frac{(x+9)^2 \cdot x^2}{x \cdot 2(x+9)}$$ 6. **Step 4: Simplify by canceling common factors:** Cancel one factor of $(x+9)$ from numerator and denominator: $$\frac{(x+9)^{\cancel{2}} \cdot x^2}{x \cdot 2 (x+9)^{\cancel{1}}} = \frac{(x+9) \cdot x^2}{2x}$$ Cancel one factor of $x$: $$\frac{(x+9) \cdot x^{\cancel{2}}}{2 x^{\cancel{1}}} = \frac{(x+9) \cdot x}{2}$$ 7. **Final simplified expression:** $$\frac{x(x+9)}{2}$$ This is the simplified form of the given expression.