To find the points of intersection, set the equations equal:

\(2x^2 + m(2x + 1) = 6x + 4\).

Rearrange to form a quadratic equation:

\(2x^2 + 2mx + m - 6x - 4 = 0\).

Simplify to:

\(2x^2 + (2m - 6)x + (m - 4) = 0\).

Use the discriminant \(b^2 - 4ac\) to determine the nature of the roots. Here, \(a = 2\), \(b = 2m - 6\), \(c = m - 4\).

The discriminant is:

\((2m - 6)^2 - 4(2)(m - 4)\).

Calculate:

\(4m^2 - 24m + 36 - 8m + 32\).

Simplify to:

\(4m^2 - 32m + 68\).

Factor or complete the square:

\((2m - 8)^2 + 1\).

This expression is always positive, indicating two distinct real roots for all values of \(m\).

Thus, the line intersects the curve at two distinct points for all values of \(m\).