Answer: \(k\lt 0\) or \(k\gt 1\).

The key point is to use the discriminant condition for the required number of real roots.

For the equation to have two different real roots, the discriminant must be positive.

Here

\(a=k,\qquad b=4k,\qquad c=3k+1.\)

So the discriminant is

\(b^2-4ac=(4k)^2-4k(3k+1).\)

Simplifying,

\(16k^2-12k^2-4k=4k^2-4k.\)

Therefore

\(4k^2-4k\gt 0.\)

Factorise:

\(4k(k-1)\gt 0.\)

This product is positive when both factors are positive or both are negative. Hence

\(k\gt 1\quad\text{or}\quad k\lt 0.\)