Can we show these two cubic curves touch?

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A curve is given by the equation \[\begin{equation*} y = ax^3 - 6ax^2 + (12a + 12)x - (8a + 16), \label{eq:1}\tag{$*$} \end{equation*}\] where \(a\) is a real number. Show that this curve touches the curve with equation \[\begin{equation*} y = x^3 \label{eq:2}\tag{$*{*}$} \end{equation*}\]

at \((2,8)\). Determine the coordinates of any other point of intersection of the two curves.

1. Sketch on the same axes the curves \(\eqref{eq:1}\) and \(\eqref{eq:2}\) when \(a = 2\).

2. Sketch on the same axes the curves \(\eqref{eq:1}\) and \(\eqref{eq:2}\) when \(a = 1\).

3. Sketch on the same axes the curves \(\eqref{eq:1}\) and \(\eqref{eq:2}\) when \(a = -2\).