4. The curve \(y = \sqrt { 2 x - 1 }\) is stretched by scale factor \(\frac { 1 } { 4 }\) parallel to the \(x\)-axis and by scale factor \(\frac { 1 } { 2 }\) parallel to the \(y\)-axis. Find the resulting equation of the curve, giving your answer in the form \(\sqrt { a x - b }\) where \(a\) and \(b\) are rational numbers.
[0pt] [BLANK PAGE] \section*{5. 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.
   [0pt] [BLANK PAGE]