The complex number \(-1 + \sqrt{7}i\) is denoted by \(u\). It is given that \(u\) is a root of the equation

\(2x^3 + 3x^2 + 14x + k = 0,\)

where \(k\) is a real constant.

(a) Find the value of \(k\). [3]

(b) Find the other two roots of the equation. [4]

(c) On an Argand diagram, sketch the locus of points representing complex numbers \(z\) satisfying the equation \(|z - u| = 2\). [2]

(d) Determine the greatest value of \(\arg z\) for points on this locus, giving your answer in radians. [2]