1. **State the problem:** Simplify the expression $$\frac{3x}{x-2} \times \frac{x}{2x+4} \times \frac{5}{x}$$. 2. **Recall the rules:** When multiplying fractions, multiply the numerators together and the denominators together. 3. **Write the expression as a single fraction:** $$\frac{3x \times x \times 5}{(x-2)(2x+4) \times x}$$ 4. **Simplify the numerator:** $$3x \times x \times 5 = 15x^2$$ 5. **Factor the denominator:** Note that $$2x+4 = 2(x+2)$$, so denominator is: $$(x-2) \times 2(x+2) \times x = 2x(x-2)(x+2)$$ 6. **Rewrite the expression:** $$\frac{15x^2}{2x(x-2)(x+2)}$$ 7. **Cancel common factors:** Cancel one $$x$$ from numerator and denominator: $$\frac{15\cancel{x}x}{2\cancel{x}(x-2)(x+2)} = \frac{15x}{2(x-2)(x+2)}$$ 8. **Final simplified expression:** $$\boxed{\frac{15x}{2(x-2)(x+2)}}$$ This is the simplified form of the original expression.