If \(x - 4\) is a factor of \(x^2 - x - k\), it means that when you divide \(x^2 - x - k\) by \(x - 4\), the remainder is zero. In other words, if you substitute \(x = 4\) into the polynomial \(x^2 - x - k\), it should result in zero, since the factor \(x - 4\) cancels out.

Let's substitute \(x = 4\) into \(x^2 - x - k\):

\(4^2 - 4 - k = 16 - 4 - k = 12 - k\)

For the polynomial to be zero, \(12 - k\) must be equal to zero:

12 - k = 0

Solving for \(k\):

k = 12