To find the values of k for which the line and the curve intersect at two distinct points, we equate the two equations:

\(2x - k = x^2 + kx - 2\)

Rearrange to form a quadratic equation:

\(x^2 + (k - 2)x + (k - 2) = 0\)

For the quadratic to have two distinct solutions, the discriminant must be greater than zero:

\(b^2 - 4ac > 0\)

Here, \(a = 1\), \(b = k - 2\), and \(c = k - 2\). So,

\((k - 2)^2 - 4(k - 2) > 0\)

Simplify:

\((k - 2)^2 - 4(k - 2) = (k - 2)(k - 6) > 0\)

This inequality holds when:

\(k < 2 \text{ or } k > 6\)