To find the values of \(m\) for which the line and the curve do not meet, we set the equations equal to each other:

\(mx - 3 = 2x^2 + 5\)

Rearrange to form a quadratic equation:

\(2x^2 - mx + 8 = 0\)

For the line and the curve not to meet, the quadratic equation must have no real solutions. This occurs when the discriminant \(b^2 - 4ac\) is less than zero.

Here, \(a = 2\), \(b = -m\), and \(c = 8\).

Calculate the discriminant:

\((-m)^2 - 4(2)(8) = m^2 - 64\)

Set the discriminant less than zero:

\(m^2 - 64 < 0\)

\(m^2 < 64\)

Taking the square root of both sides gives:

\(-8 < m < 8\)