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 \(m\).