Answer: \(6\lt k\lt10\).

We start with the main method. Use the discriminant to decide how many real roots the quadratic equation has.

For the curve

\(y=x^2+kx+4k-15\)

to be completely above the \(x\)-axis, the quadratic must have no real roots and open upwards.

Since the coefficient of \(x^2\) is \(1\), it opens upwards.

So we need the discriminant to be negative:

\(k^2-4(1)(4k-15)\lt0.\)

This gives

\(k^2-16k+60\lt0.\)

Factorising,

\((k-6)(k-10)\lt0.\)

Therefore \(k\) lies between the two roots:

\(6\lt k\lt10.\)

This completes the solution and gives the required result.