The equation \(2 x ^ { 3 } - x ^ { 2 } + 2 x + 12 = 0\) has one real root and two complex roots. Showing your working, verify that \(1 + i \sqrt { } 3\) is one of the complex roots. State the other complex root.
On a sketch of an Argand diagram, show the point representing the complex number \(1 + \mathrm { i } \sqrt { } 3\). On the same diagram, shade the region whose points represent the complex numbers \(z\) which satisfy both the inequalities \(| z - 1 - i \sqrt { } 3 | \leqslant 1\) and \(\arg z \leqslant \frac { 1 } { 3 } \pi\).
Express \(\frac { 4 + 5 x - x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }\) in partial fractions.
Hence obtain the expansion of \(\frac { 4 + 5 x - x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).