The complex number 1 + 2i is denoted by u. The polynomial 2x^3 + ax^2 + 4x + b, where a and b are real constants, is denoted by p(x). It is given that u is a root of the equation p(x) = 0.

(a) Find the values of a and b.

(b) State a second complex root of this equation.

(c) Find the real factors of p(x).

(d) (i) On a sketch of an Argand diagram, shade the region whose points represent complex numbers z satisfying the inequalities |z - u| ≤ √5 and arg z ≤ 1/4 π.

(ii) Find the least value of Im z for points in the shaded region. Give your answer in an exact form.