1. Problem: Find the roots of the quadratic equation $x^2 + 4x + 3 = 0$ by completing the square. 2. Formula: To complete the square for $x^2 + bx + c = 0$, rewrite as $\left(x + \frac{b}{2}\right)^2 = \left(\frac{b}{2}\right)^2 - c$. 3. Apply to $x^2 + 4x + 3 = 0$: $$x^2 + 4x + 3 = 0$$ Move constant term: $$x^2 + 4x = -3$$ Add $\left(\frac{4}{2}\right)^2 = 4$ to both sides: $$x^2 + 4x + 4 = -3 + 4$$ Intermediate step showing cancellation: $$x^2 + \cancel{4x} + \cancel{4} = -3 + 4$$ Rewrite left side as a perfect square: $$\left(x + 2\right)^2 = 1$$ 4. Solve for $x$: $$x + 2 = \pm 1$$ So, $$x = -2 \pm 1$$ 5. Final roots: $$x = -1 \text{ or } x = -3$$ This completes the solution for the first quadratic equation by completing the square.