Answer: \(-\frac52\leq k\leq2\).

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

For real roots, the discriminant must be non-negative:

\(b^2-4ac\geq0.\)

Here

\(a=2k-1,\qquad b=6,\qquad c=k+1.\)

So

\(6^2-4(2k-1)(k+1)\geq0.\)

Expand:

\(36-4(2k^2+k-1)\geq0.\)

Therefore

\(40-4k-8k^2\geq0.\)

Divide by \(-4\), reversing the inequality:

\(2k^2+k-10\leq0.\)

Factorise:

\((2k+5)(k-2)\leq0.\)

Hence

\(-\frac52\leq k\leq2.\)