1. **State the problem:** Solve the quadratic equation $x^2 + 8x + 11 = 0$ by completing the square. 2. **Recall the formula and method:** To complete the square for an equation of the form $x^2 + bx + c = 0$, we rewrite it as $(x + d)^2 = e$ where $d = \frac{b}{2}$ and then solve for $x$. 3. **Rewrite the equation:** $$x^2 + 8x + 11 = 0$$ Move the constant term to the right side: $$x^2 + 8x = -11$$ 4. **Complete the square:** Take half of the coefficient of $x$, which is $\frac{8}{2} = 4$, and square it: $$4^2 = 16$$ Add 16 to both sides to keep the equation balanced: $$x^2 + 8x + 16 = -11 + 16$$ $$ (x + 4)^2 = 5$$ 5. **Solve for $x$:** Take the square root of both sides: $$x + 4 = \pm \sqrt{5}$$ Subtract 4 from both sides: $$x = -4 \pm \sqrt{5}$$ 6. **Final answer:** $$x = -4 + \sqrt{5} \quad \text{or} \quad x = -4 - \sqrt{5}$$ This method transforms the quadratic into a perfect square trinomial, making it easier to solve by taking square roots.