1. **State the problem:** Solve the quadratic equation $x^2 + 4x + 3 = 0$. 2. **Formula and rules:** The quadratic equation is generally solved using the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a$, $b$, and $c$ are coefficients from the equation $ax^2 + bx + c = 0$. 3. **Identify coefficients:** Here, $a = 1$, $b = 4$, and $c = 3$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 4^2 - 4 \times 1 \times 3 = 16 - 12 = 4$$ 5. **Apply the quadratic formula:** $$x = \frac{-4 \pm \sqrt{4}}{2 \times 1} = \frac{-4 \pm 2}{2}$$ 6. **Find the two solutions:** - For the plus sign: $$x = \frac{-4 + 2}{2} = \frac{-2}{2} = -1$$ - For the minus sign: $$x = \frac{-4 - 2}{2} = \frac{-6}{2} = -3$$ 7. **Final answer:** The solutions to the equation are $x = -1$ and $x = -3$.