To complete the square for the quadratic equation $x^2 + 6x + 11 = 0$, follow these steps:

1. Start with the equation: $x^2 + 6x + 11 = 0$

2. Move the constant to the other side: $x^2 + 6x = -11$

3. Complete the square by adding $\left(\frac{6}{2}\right)^2 = 9$ to both sides: $x^2 + 6x + 9 = -11 + 9$ $(x + 3)^2 = -2$

4. Take the square root of both sides: $x + 3 = \pm \sqrt{-2}$ $x + 3 = \pm i\sqrt{2}$

5. Solve for $x$: $x = -3 \pm i\sqrt{2}$

So, the solutions are: $x = -3 + i\sqrt{2} \quad \text{or} \quad x = -3 - i\sqrt{2}$

Would you like more details on any step, or do you have questions?

Here are five related questions for further exploration:

1. How do you complete the square for any quadratic equation?
2. What is the geometric interpretation of completing the square?
3. How can you solve a quadratic equation using the quadratic formula?
4. What are complex numbers, and how do they arise in equations like this?
5. How do real and complex solutions of quadratics differ?

Tip: Completing the square is not only useful for solving quadratics but also for understanding the vertex form of a parabola.