10. The curve \(C\) with equation \(y = \mathrm { f } ( x )\) is such that
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } + 4 x + k$$
where \(k\) is a constant.
Given that \(C\) passes through the points \(( 0 , - 2 )\) and \(( 2,18 )\),
- show that \(k = 2\) and find an equation for \(C\),
- show that the line with equation \(y = x - 2\) is a tangent to \(C\) and find the coordinates of the point of contact.