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 \(c > 3\).

To find the set of values of \(k\) for which the curve and line do not intersect, we set the equations equal to each other:

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

Rearrange to form a quadratic equation:

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

For the curve and line not to intersect, the discriminant of this quadratic must be less than zero:

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

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

Calculate the discriminant:

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

\(4 - 4k + k^2 - 8k + 4k^2 < 0\)

\(5k^2 - 12k + 4 < 0\)

Factor the quadratic:

\((k-2)(5k-2) < 0\)

Find the critical points by setting each factor to zero:

\(k - 2 = 0 \Rightarrow k = 2\)

\(5k - 2 = 0 \Rightarrow k = \frac{2}{5}\)

The inequality \((k-2)(5k-2) < 0\) holds between the roots:

\(\frac{2}{5} < k < 2\)

For the quadratic equation \(ax^2 + bx + c = 0\) to have exactly one root, the discriminant must be zero. The discriminant \(\Delta\) is given by \(b^2 - 4ac\).

Here, \(a = 16\), \(b = -24\), and \(c = 10 - k\).

Set the discriminant to zero:

\((-24)^2 - 4 \times 16 \times (10 - k) = 0\)

\(576 - 64(10 - k) = 0\)

\(576 - 640 + 64k = 0\)

\(-64 + 64k = 0\)

\(64k = 64\)

\(k = 1\)

Substitute \(k = 1\) back into the equation:

\(16x^2 - 24x + 9 = 0\)

Factor the quadratic:

\((4x - 3)^2 = 0\)

Thus, \(4x - 3 = 0\)

\(4x = 3\)

\(x = \frac{3}{4}\)