Let \(\mathrm{h}(z) = nz^6 + z^5 + z + n\), where \(z\) is a complex number and \(n \geqslant 2\) is an integer.
Let \(w\) be a root of the equation \(\mathrm{h}(z) = 0\).
Show that \(|w^5| = \sqrt{\dfrac{\mathrm{f}(w)}{\mathrm{g}(w)}}\), where
\[\mathrm{f}(z) = n^2 + 2n\operatorname{Re}(z) + |z|^2 \quad \text{and} \quad \mathrm{g}(z) = n^2|z|^2 + 2n\operatorname{Re}(z) + 1.\]
By considering \(\mathrm{f}(w) - \mathrm{g}(w)\), prove by contradiction that \(|w| \geqslant 1\).
Show that \(|w| = 1\).
It is given that the equation \(\mathrm{h}(z) = 0\) has six distinct roots, none of which is purely real.
Show that \(\mathrm{h}(z)\) can be written in the form
\[\mathrm{h}(z) = n(z^2 - a_1 z + 1)(z^2 - a_2 z + 1)(z^2 - a_3 z + 1),\]
where \(a_1\), \(a_2\) and \(a_3\) are real constants.
Find \(a_1 + a_2 + a_3\) in terms of \(n\).
By considering the coefficient of \(z^3\) in \(\mathrm{h}(z)\), find \(a_1 a_2 a_3\) in terms of \(n\).
How many of the six roots of the equation \(\mathrm{h}(z) = 0\) have a negative real part? Justify your answer.