Solve the Quadratic Puzzle: Factoring x² - 64

To solve this problem, we need to recognize the expression $x^2 - 64$ as a difference of squares.

The difference of squares formula states: $a^2 - b^2 = (a - b)(a + b)$.

In this problem, we identify that:

• $a = x$

• $b^2 = 64$, which means $b = \sqrt{64} = 8$

Therefore, applying the formula gives us:

$x^2 - 64 = (x - 8)(x + 8)$

This indicates that the missing element in the expression $(x - \_)(\_ + x)$ is $8$.

Thus, the correct answer to fill in the missing element is $\boxed{8}$, corresponding to choice 4.