1. **State the problem:** Factor the cubic polynomial $$x^3 - 15x^2 + 50x$$ completely. 2. **Identify common factors:** Notice that each term contains an $$x$$, so factor $$x$$ out first: $$x^3 - 15x^2 + 50x = x(x^2 - 15x + 50)$$ 3. **Factor the quadratic:** Now factor $$x^2 - 15x + 50$$. We look for two numbers that multiply to $$50$$ and add to $$-15$$. The factors of $$50$$ are $$-5$$ and $$-10$$ because $$-5 \times -10 = 50$$ and $$-5 + -10 = -15$$. 4. **Write the factorization:** $$x^2 - 15x + 50 = (x - 5)(x - 10)$$ 5. **Combine all factors:** $$x^3 - 15x^2 + 50x = x(x - 5)(x - 10)$$ **Final answer:** $$x(x - 5)(x - 10)$$