7 The curve \(C\) has equation \(y = k \left( x ^ { 2 } + 3 \right)\), where \(k\) is a constant.
The line \(L\) has equation \(y = 2 x + 2\).
- Show that the \(x\)-coordinates of any points of intersection of the curve \(C\) with the line \(L\) satisfy the equation
$$k x ^ { 2 } - 2 x + 3 k - 2 = 0$$
- The curve \(C\) and the line \(L\) intersect in two distinct points.
- Show that
$$3 k ^ { 2 } - 2 k - 1 < 0$$
- Hence find the possible values of \(k\).