Answer:
2
Step-by-step explanation:
To determine the value of \( k \) for which \( x + k \) is a factor of \( x^2 + kx + k - 2 \), we proceed as follows:
Since \( x + k \) is a factor, the polynomial \( x^2 + kx + k - 2 \) must be divisible by \( x + k \). By the factor theorem, if \( x + k \) is a factor, then \( x = -k \) is a root of the polynomial.
Substitute \( x = -k \) into \( x^2 + kx + k - 2 \):
\[
(-k)^2 + k(-k) + k - 2 = k^2 - k^2 + k - 2 = k - 2
\]
For \( x + k \) to be a factor, \( k - 2 \) must equal \( 0 \) (since the remainder when \( x^2 + kx + k - 2 \) is divided by \( x + k \) should be zero):
\[
k - 2 = 0
\]
\[
k = 2
\]
Therefore, the value of \( k \) for which \( x + k \) is a factor of \( x^2 + kx + k - 2 \) is \( \boxed{2} \).