1. **State the problem:** Solve the quadratic equation $$x^2 + 19x = 0$$. 2. **Formula and rules:** To solve quadratic equations set equal to zero, factor the expression and use the zero product property: if $$ab=0$$, then $$a=0$$ or $$b=0$$. 3. **Factor the equation:** $$x^2 + 19x = x(x + 19) = 0$$ 4. **Apply zero product property:** $$x = 0 \quad \text{or} \quad x + 19 = 0$$ 5. **Solve each equation:** $$x = 0$$ $$x = -19$$ 6. **Final answer:** The solution set is $$\{0, -19\}$$.