Answer: \(-4\lt k\lt 20\).

At an intersection, the \(y\)-values are equal:

\(kx-7=3x^2+8x+5.\)

Rearrange:

\(3x^2+(8-k)x+12=0.\)

The line and curve do not intersect when this quadratic has no real roots.

So its discriminant must be less than zero:

\((8-k)^2-4(3)(12)\lt 0.\)

Thus

\((8-k)^2-144\lt 0.\)

So

\((8-k)^2\lt 144.\)

Therefore

\(-12\lt 8-k\lt 12.\)

Subtracting 8 gives

\(-20\lt -k\lt 4.\)

Multiplying by \(-1\), and reversing the inequalities, gives

\(-4\lt k\lt 20.\)