Let's solve the quadratic equation by completing the square:

The equation is:

$x^2 + 6x + 8 = 0$

Step 1: Move the constant term to the other side

$x^2 + 6x = -8$

Step 2: Complete the square

To complete the square, take half of the coefficient of $x$, which is 6, divide by 2, and then square it:

$\left(\frac{6}{2}\right)^2 = 3^2 = 9$

Add 9 to both sides of the equation:

$x^2 + 6x + 9 = -8 + 9$

This simplifies to:

$(x + 3)^2 = 1$

Step 3: Solve for $x$

Take the square root of both sides:

$x + 3 = \pm 1$

Step 4: Solve for $x$

Now, solve for $x$ by isolating it:

1. $x + 3 = 1 \Rightarrow x = 1 - 3 = -2$
2. $x + 3 = -1 \Rightarrow x = -1 - 3 = -4$

Final solution:

$x = \{-2, -4\}$

Would you like any additional details or have any further questions?

Here are some related questions:

1. What is the general form of completing the square?
2. How do you handle coefficients in front of $x^2$ when completing the square?
3. Can completing the square be used to solve any quadratic equation?
4. What are the advantages of completing the square over the quadratic formula?
5. How is completing the square related to the vertex form of a quadratic?

Tip: Completing the square is also useful for converting quadratic equations to vertex form, which reveals important properties like the vertex of the parabola.