Answer: \(k=-\frac34\) or \(k=1\).

For a quadratic equation to have two equal roots, its discriminant must be zero.

The equation is

\(x^2+4kx+k+3=0.\)

Here

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

So the discriminant condition is

\((4k)^2-4(1)(k+3)=0.\)

Simplify:

\(16k^2-4k-12=0.\)

Divide by \(4\):

\(4k^2-k-3=0.\)

Factorise:

\((4k+3)(k-1)=0.\)

Therefore

\(k=-\frac34\quad\text{or}\quad k=1.\)