8. The curve \(C _ { 1 }\) has equation
$$y = 3 x ^ { 2 } + 6 x + 9$$
- Write \(3 x ^ { 2 } + 6 x + 9\) in the form
$$a ( x + b ) ^ { 2 } + c$$
where \(a\), \(b\) and \(c\) are constants to be found.
The point \(P\) is the minimum point of \(C _ { 1 }\)
- Deduce the coordinates of \(P\).
A different curve \(C _ { 2 }\) has equation
$$y = A x ^ { 3 } + B x ^ { 2 } + C x + D$$
where \(A\), \(B\), \(C\) and \(D\) are constants.
Given that \(C _ { 2 }\)
- passes through \(P\)
- intersects the \(x\)-axis at \(- 4 , - 2\) and 3
- find, making your method clear, the values of \(A , B , C\) and \(D\).
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