2 In this question you must show detailed reasoning.
You are given that $\mathrm { f } ( z ) = 4 z ^ { 4 } - 12 z ^ { 3 } + 41 z ^ { 2 } - 128 z + 185$ and that $2 + \mathrm { i }$ is a root of the equation $\mathrm { f } ( z ) = 0$.
\begin{enumerate}[label=(\alph*)]
\item Express $\mathrm { f } ( z )$ as the product of two quadratic factors with integer coefficients.
\item Solve $\mathrm { f } ( z ) = 0$.

Two loci on an Argand diagram are defined by $C _ { 1 } = \left\{ z : | z | = r _ { 1 } \right\}$ and $C _ { 2 } = \left\{ z : | z | = r _ { 2 } \right\}$ where $r _ { 1 } > r _ { 2 }$. You are given that two of the points representing the roots of $\mathrm { f } ( z ) = 0$ are on $C _ { 1 }$ and two are on $C _ { 2 } \cdot R$ is the region on the Argand diagram between $C _ { 1 }$ and $C _ { 2 }$.
\item Find the exact area of $R$.
\item $\omega$ is the sum of all the roots of $\mathrm { f } ( z ) = 0$.

Determine whether or not the point on the Argand diagram which represents $\omega$ lies in $R$.
\end{enumerate}

\hfill \mbox{\textit{OCR FP1 AS 2021 Q2 [14]}}