8. A circle, \(C\), has equation \(x ^ { 2 } - 6 x + y ^ { 2 } = 16\).
A second circle, \(D\), has the following properties:

• The line through the centres of circle \(C\) and circle \(D\) has gradient 1 .
• Circle \(D\) touches circle \(C\) at exactly one point.
• The centre of circle \(D\) lies in the first quadrant.
• Circle \(D\) has the same radius as circle \(C\).

Find the coordinates of the centre of circle \(D\). \section*{9. In this question you must show detailed reasoning.} The polynomial \(\mathrm { f } ( x )\) is given by $$f ( x ) = x ^ { 3 } + 6 x ^ { 2 } + x - 4$$

1. (a) Show that \(( x + 1 )\) is a factor of \(\mathrm { f } ( x )\).
   (b) Hence find the exact roots of the equation \(\mathrm { f } ( x ) = 0\).
2. (a) Show that the equation $$2 \log _ { 2 } ( x + 3 ) + \log _ { 2 } x - \log _ { 2 } ( 4 x + 2 ) = 1$$ can be written in the form \(\mathrm { f } ( x ) = 0\).
   (b) Explain why the equation $$2 \log _ { 2 } ( x + 3 ) + \log _ { 2 } x - \log _ { 2 } ( 4 x + 2 ) = 1$$ has only one real root and state the exact value of this root.