To find the values of k for which the line intersects the curve at two distinct points, we set the equations equal to each other:

\(kx - 4 = x^2 - 2x\)

Rearrange to form a quadratic equation:

\(x^2 - (2 + k)x + 4 = 0\)

For the line to intersect the curve at two distinct points, the discriminant of this quadratic equation must be greater than zero. The discriminant \(\Delta\) is given by:

\(\Delta = b^2 - 4ac\)

Here, \(a = 1\), \(b = -(2 + k)\), and \(c = 4\). Thus,

\(\Delta = (2 + k)^2 - 4 \times 1 \times 4\)

\(\Delta = (2 + k)^2 - 16\)

For two distinct points, \(\Delta > 0\):

\((2 + k)^2 > 16\)

Solving this inequality:

\(2 + k > 4\) or \(2 + k < -4\)

\(k > 2\) or \(k < -6\)