To show that \(4c = m^2 - 12m + 16\), we start by equating the line and the curve: \(mx + c = 6x - x^2 - 5\).

Rearrange to form a quadratic equation: \(x^2 - (6-m)x + (c+5) = 0\).

For the line to be tangent to the curve, the discriminant of this quadratic must be zero: \(b^2 - 4ac = 0\).

Here, \(a = 1\), \(b = -(6-m)\), and \(c = c+5\).

Calculate the discriminant: \((6-m)^2 - 4(1)(c+5) = 0\).

Expand and simplify: \(36 - 12m + m^2 - 4c - 20 = 0\).

Rearrange to find \(4c = m^2 - 12m + 16\).