I am writing as follows:
The third-order polynomial of a complex variable $x,m,a,b \in \Bbb C$ is written as,
$m = ax + b{\left| x \right|^2}x$
Where, ${\left| x \right|^2} = x{x^*}$ represent norm 2 square of $x$, and $x^*$ is the conjugate of $x$.
Let, $x = cy + d{\left| y \right|^2}y$; where $c,d,y \in \Bbb C$.
Therefore,
$m = a\left( {cy + d{{\left| y \right|}^2}y} \right) + b{\left| {cy + d{{\left| y \right|}^2}y} \right|^2}\left( {cy + d{{\left| y \right|}^2}y} \right)$
Now, my complicacy is to expand ${\left| {cy + d{{\left| y \right|}^2}y} \right|^2}$.
\begin{align} {\left| {cy + d{{\left| y \right|}^2}y} \right|^2} &= \left( {cy + d{{\left| y \right|}^2}y} \right){\left( {cy + d{{\left| y \right|}^2}y} \right)^*}\\ &= \left( {cy + d{{\left| y \right|}^2}y} \right)\left( {c^*y^* + d^*{{\left| y \right|}^2}y^*} \right)\\ &= {\left| c \right|^2}{\left| y \right|^2} + cd^*{\left| y \right|^4} + dc^*{\left| y \right|^4} + {\left| d \right|^2}{\left| y \right|^6} \end{align}
Am I on the right track? Or did I do any mistake?
Any suggestions will help. Thanks.