(a) Substitute \(a = 2 + yi\) into \(f(a) = a^3 - a^2 - 2a\).

Calculate \(a^2 = (2 + yi)^2 = 4 + 4yi - y^2i^2 = 4 + 4yi + y^2\).

Calculate \(a^3 = (2 + yi)^3 = 8 + 12yi - 6y^2 - y^3i\).

Substitute into \(f(a) = a^3 - a^2 - 2a\):

\(f(a) = (8 + 12yi - 6y^2 - y^3i) - (4 + 4yi + y^2) - 2(2 + yi)\).

Simplify to get \(-5y^2 + (6y - y^3)i\).

(b) Given \(\text{Re}(f(a)) = -20\), equate \(-5y^2 = -20\) to solve for \(y\).

\(-5y^2 = -20 \Rightarrow y^2 = 4 \Rightarrow y = -2\) (since \(y < 0\)).

Now, \(a = 2 - 2i\). The argument \(\arg a = \arctan\left(\frac{-2}{2}\right) = \arctan(-1) = -\frac{\pi}{4}\).