Hopefully this is right....
Im(2 + z +3z^4)
let, z= rcosX+isinX
z^n = r^n {cos(nX)+isin(nX)} by de moivre
Im(2 + z +3z^4) = Im(2 + r{cosX+isinX} + 3r^4 cos(4X)+isin(4X))
= rsinX + 3r^4(sin{4X})
but |z|<=1, so r<=1 for all X
so Im(2 + z +3z^4) <= 1 + 3 = 4
QED