If both the roots of the quadratic equation $x^2-2kx+k^2+k-5=0$ are less than 5, then $k$  lies in the interval

$f\left(5\right)>0,\Delta >0$

$25-10k+k^2+k-5>0,4k^2-4(k^2+k-5)>0$

$k^2-9k+20>0 k-5<0$

$(k-4)(k-5)>0 k<5......\left(1\right)$

$k<4ork>5..........\left(2\right)$

From (1) and (2) k < 4

$k\in (-\infty ,4)$