To find the values of k for which the line y = 4x + k does not intersect the curve y = x^2, we set the equations equal to each other:

\(x^2 = 4x + k\)

Rearrange to form a quadratic equation:

\(x^2 - 4x - k = 0\)

For the line not to intersect the curve, the quadratic equation must have no real solutions. This occurs when the discriminant is less than zero:

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

Here, a = 1, b = -4, and c = -k. Substitute these into the discriminant:

\((-4)^2 - 4(1)(-k) < 0\)

Simplify:

\(16 + 4k < 0\)

Solve for k:

\(4k < -16\)

\(k < -4\)

Thus, the set of values for k is k < -4.