1. **Statement of the problem:** Solve the quadratic equation $$x^2 + 8x = 0$$. 2. **Rewrite the equation:** The equation can be factored by taking $$x$$ common: $$x^2 + 8x = x(x + 8) = 0$$ 3. **Zero product property:** If a product of two factors is zero, then at least one of the factors must be zero. So, $$x = 0 \quad \text{or} \quad x + 8 = 0$$ 4. **Solve each factor:** - For $$x = 0$$, the solution is $$x_1 = 0$$. - For $$x + 8 = 0$$, subtract 8 from both sides: $$x + 8 = 0 \Rightarrow x = -8$$ 5. **Final solutions:** $$x_1 = 0, \quad x_2 = -8$$ These are the roots of the quadratic equation.