2025 DHS Promo Q8

A curve \(C\) has equation \(\mathrm{f}(x)=\dfrac{3x^2+6x+k}{x+2}\), where \(x>-2\) and \(k\) is a non-zero real constant.

It is given that the function \(\mathrm{f}\) is strictly increasing.

1. Find the range of values of \(k\).[3]
2. Sketch the curve \(C\), giving the coordinates of any points where \(C\) crosses the \(x\)- and \(y\)-axes and the equations of any asymptotes.[3]
3. By adding a suitable curve to the graph of the curve \(C\) in (b), deduce the number of distinct real root(s) of the equation \((3x^2+6x+k)^2=(x^2+1)(9x^2+36x+36)\).[3]