Answer: \(k\lt 2-2\sqrt3\) or \(k\gt 2+2\sqrt3\).

At a point of intersection, the two expressions for \(y\) are equal:

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

Rearrange this into a quadratic equation in \(x\):

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

The line meets the curve at two distinct points exactly when this quadratic has two distinct real roots. Therefore its discriminant must be positive:

\((2-k)^2-4(1)(3)\gt0.\)

So

\((2-k)^2\gt12.\)

Taking square roots gives

\(|2-k|\gt2\sqrt3.\)

This means either

\(2-k\gt2\sqrt3\quad\text{or}\quad 2-k\lt-2\sqrt3.\)

Hence

\(k\lt2-2\sqrt3\quad\text{or}\quad k\gt2+2\sqrt3.\)