To solve the quadratic equation $x^2 + 8x - 5 = 0$ by completing the square, follow these steps:

1. Move the constant term to the other side:

$x^2 + 8x = 5$

2. Complete the square:

To complete the square, take half of the coefficient of $x$ (which is 8), divide it by 2, and then square it. This value will be added to both sides.

$\left(\frac{8}{2}\right)^2 = 16$

Now, add 16 to both sides of the equation: $x^2 + 8x + 16 = 5 + 16$

$(x + 4)^2 = 21$

3. Solve for $x$:

Take the square root of both sides: $x + 4 = \pm \sqrt{21}$

4. Isolate $x$:

$x = -4 \pm \sqrt{21}$

Thus, the two solutions are: $x = -4 + \sqrt{21} \quad \text{or} \quad x = -4 - \sqrt{21}$

Would you like more details on any specific part of the process or further explanation?

Here are 5 questions to extend your understanding:

1. How can completing the square help with graphing the parabola of a quadratic function?
2. What are some other methods to solve quadratic equations besides completing the square?
3. What does the discriminant tell you about the number and type of solutions in a quadratic equation?
4. How would the solution change if the equation had a different constant term, say $x^2 + 8x + 3 = 0$?
5. What does the term $x = -4 \pm \sqrt{21}$ mean geometrically?

Tip: Completing the square is also useful for converting quadratic equations into vertex form, which makes it easier to analyze the graph of a quadratic function.