To find the value of \(c\) where the line is tangent to the curve, we need to set the equations equal and solve for \(x\) and \(c\).

First, equate the line and curve equations:

\(2x + c = 8 - 2x - x^2\)

Rearrange to form a quadratic equation:

\(x^2 + 4x + c - 8 = 0\)

For the line to be tangent to the curve, the discriminant of this quadratic must be zero:

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

Here, \(a = 1\), \(b = 4\), and \(c = c - 8\).

Substitute into the discriminant formula:

\(4^2 - 4(1)(c - 8) = 0\)

Simplify:

\(16 - 4c + 32 = 0\)

\(48 - 4c = 0\)

Solve for \(c\):

\(4c = 48\)

\(c = 12\)