To find the intersection points, equate the equations of the curve and the line:

\(x^2 + 2cx + 4 = 4x + c\)

Rearrange to form a quadratic equation:

\(x^2 + 2cx - 4x + 4 - c = 0\)

This simplifies to:

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

For the curve and line to intersect at two distinct points, the discriminant \(b^2 - 4ac\) must be greater than zero.

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

Calculate the discriminant:

\((2c - 4)^2 - 4(1)(4 - c) > 0\)

Expand and simplify:

\(4c^2 - 16c + 16 - 16 + 4c > 0\)

\(4c^2 - 12c > 0\)

Factorize:

\(4c(c - 3) > 0\)

This inequality holds when \(c < 0\) or \(c > 3\).