1. **State the problem:** Solve the quadratic equation $2x^2 + 3x + 2 = -3$. 2. **Rewrite the equation:** Move all terms to one side to set the equation equal to zero: $$2x^2 + 3x + 2 + 3 = 0$$ which simplifies to $$2x^2 + 3x + 5 = 0$$ 3. **Identify coefficients:** Here, $a = 2$, $b = 3$, and $c = 5$. 4. **Use the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 5. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 3^2 - 4 \times 2 \times 5 = 9 - 40 = -31$$ 6. **Interpret the discriminant:** Since $\Delta < 0$, the equation has no real solutions but two complex solutions. 7. **Find the complex solutions:** $$x = \frac{-3 \pm \sqrt{-31}}{2 \times 2} = \frac{-3 \pm i\sqrt{31}}{4}$$ 8. **Final answer:** $$x = \frac{-3}{4} + \frac{i\sqrt{31}}{4} \quad \text{or} \quad x = \frac{-3}{4} - \frac{i\sqrt{31}}{4}$$