To find the points of intersection, set the equations equal: \(3x + k = x^2 + kx + 6\).

Rearrange to form a quadratic equation: \(x^2 + (k - 3)x + (6 - k) = 0\).

For two distinct points of intersection, the discriminant must be greater than zero: \(b^2 - 4ac > 0\).

Here, \(a = 1\), \(b = k - 3\), and \(c = 6 - k\).

Calculate the discriminant: \((k - 3)^2 - 4(6 - k) > 0\).

Simplify: \(k^2 - 2k - 15 > 0\).

Factorize: \((k + 3)(k - 5) > 0\).

The solution to this inequality is \(k < -3\) or \(k > 5\).