Step 1 :Let's substitute \(x = -4\) into the equation \(q(x) = -40 - 2x + 2x^{2}\).
Step 2 :After substituting, we get \(q(-4) = -40 - 2(-4) + 2(-4)^{2}\).
Step 3 :Solving the equation, we get \(q(-4) = 0\).
Step 4 :So, the final answer is \(\boxed{0}\).