N2020 P1 Q6

The complex number \(z\) satisfies the equation

\({{z}^{2}}\left( 2+\mathbf{i} \right)-8\mathbf{i}\,z+t=0\)

where \(t\) is a real number. It is given that one of the root is of the form \(k+k\,\mathbf{i}\), where \(k\) is real and non-zero. Find \(t\) and \(k\), and the other root of the equation. [8]