7. The complex numbers \(z\) and \(w\) are represented, respectively, by points \(P ( x , y )\) and \(Q ( u , v )\) in Argand diagrams and
$$w = z ( 1 + z )$$
- Show that
$$v = y ( 1 + 2 x )$$
and find an expression for \(u\) in terms of \(x\) and \(y\).
- The point \(P\) moves along the line \(y = x + 1\). Find the Cartesian equation of the locus of \(Q\), giving your answer in the form \(v = a u ^ { 2 } + b u\), where \(a\) and \(b\) are constants whose values are to be determined.