1. **State the problem:** Solve the equation $3x(x + 1) = -5$ for $x$. 2. **Rewrite the equation:** Expand the left side: $$3x(x + 1) = 3x^2 + 3x$$ So the equation becomes: $$3x^2 + 3x = -5$$ 3. **Bring all terms to one side:** $$3x^2 + 3x + 5 = 0$$ 4. **Identify coefficients:** Here, $a = 3$, $b = 3$, and $c = 5$. 5. **Use the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Substitute values: $$x = \frac{-3 \pm \sqrt{3^2 - 4 \times 3 \times 5}}{2 \times 3} = \frac{-3 \pm \sqrt{9 - 60}}{6} = \frac{-3 \pm \sqrt{-51}}{6}$$ 6. **Simplify the discriminant:** Since $\sqrt{-51} = i\sqrt{51}$ (where $i$ is the imaginary unit), the solutions are complex: $$x = \frac{-3 \pm i\sqrt{51}}{6}$$ 7. **Express final answer:** $$x = -\frac{3}{6} \pm \frac{i\sqrt{51}}{6} = -\frac{1}{2} \pm \frac{i\sqrt{51}}{6}$$ **Answer:** The solutions are complex conjugates: $$x = -\frac{1}{2} + \frac{i\sqrt{51}}{6} \quad \text{and} \quad x = -\frac{1}{2} - \frac{i\sqrt{51}}{6}$$