To find the values of m for which the line intersects the curve at two distinct points, we equate the equations of the line and the curve:

\(mx + 4 = 3x^2 - 4x + 7\)

Rearrange to form a quadratic equation:

\(3x^2 - (4 + m)x + 3 = 0\)

For the line to intersect the curve at two distinct points, the discriminant of this quadratic equation must be greater than zero:

\(b^2 - 4ac > 0\)

Here, \(a = 3\), \(b = -(4 + m)\), and \(c = 3\).

Calculate the discriminant:

\((4 + m)^2 - 4 \times 3 \times 3 > 0\)

\((4 + m)^2 - 36 > 0\)

\((4 + m)^2 > 36\)

Taking the square root of both sides:

\(|4 + m| > 6\)

This gives two inequalities:

\(4 + m > 6\) or \(4 + m < -6\)

Solving these:

\(m > 2\) or \(m < -10\)

Thus, the set of values of m for which the line intersects the curve at two distinct points is \(m > 2 \text{ or } m < -10\).