To find the points of intersection, set the equations equal: \(ax + 3a = -\frac{2}{x}\).

Multiply through by \(x\) to eliminate the fraction: \(ax^2 + 3ax + 2 = 0\).

This is a quadratic equation in \(x\): \(ax^2 + 3ax + 2 = 0\).

For the line and curve to intersect at two distinct points, the discriminant of the quadratic must be greater than zero: \(b^2 - 4ac > 0\).

Here, \(a = a\), \(b = 3a\), \(c = 2\).

Calculate the discriminant: \((3a)^2 - 4(a)(2) > 0\).

Simplify: \(9a^2 - 8a > 0\).

Factorize: \(a(9a - 8) > 0\).

The critical points are \(a = 0\) and \(a = \frac{8}{9}\).

Test intervals around the critical points to determine where the inequality holds:

- For \(a < 0\), \(a(9a - 8) > 0\) is true.

- For \(0 < a < \frac{8}{9}\), \(a(9a - 8) < 0\) is false.

- For \(a > \frac{8}{9}\), \(a(9a - 8) > 0\) is true.

Thus, the set of values for \(a\) is \(a < 0\) or \(a > \frac{8}{9}\).