The curve \(C\) has equation
\(y=\frac{p x^{2}+4 x+1}{x+1}\)
where \(p\) is a positive constant and \(p \neq 3\).
(i) Obtain the equations of the asymptotes of \(C\).

(ii) Find the value of \(p\) for which the \(x\)-axis is a tangent to \(C\), and sketch \(C\) in this case.
(iii) For the case \(p=1\), show that \(C\) has no turning points, and sketch \(C\), giving the exact coordinates of the points of intersection of \(C\) with the \(x\)-axis.