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

\(3x^2 + 2x + 4 = mx + 1\)

Rearranging gives:

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

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

\((2 - m)^2 - 4 imes 3 imes 3 > 0\)

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

\((2 - m)^2 > 36\)

Taking the square root of both sides:

\(|2 - m| > 6\)

This gives two inequalities:

\(2 - m > 6\) or \(2 - m < -6\)

Solving these inequalities:

1. \(2 - m > 6\) gives \(m < -4\)

2. \(2 - m < -6\) gives \(m > 8\)

Thus, the set of values of m for which the line and the curve intersect at two distinct points is:

\(m < -4\) or \(m > 8\)