First, find the derivative of the curve \(y = 2x^3 - 5x^2 - 3x + c\):

\(\frac{dy}{dx} = 6x^2 - 10x - 3\).

Evaluate the derivative at \(x = 2\):

\(\frac{dy}{dx} = 6(2)^2 - 10(2) - 3 = 24 - 20 - 3 = 1\).

Since the line is tangent to the curve, the slope \(a\) of the line is equal to the derivative at \(x = 2\):

\(a = 1\).

Substitute \(x = 2\) and \(y = 6\) into the line equation \(y = ax + b\):

\(6 = 2(1) + b \Rightarrow b = 4\).

Substitute \(x = 2\) and \(y = 6\) into the curve equation \(y = 2x^3 - 5x^2 - 3x + c\):

\(6 = 2(2)^3 - 5(2)^2 - 3(2) + c\).

Simplify to find \(c\):

\(6 = 16 - 20 - 6 + c \Rightarrow c = 16\).