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(A) Given the quadratic equation $f(x) = x^2 - 2kx + k^2 + k - 5 = 0$.For both roots to be less than $5$,the following conditions must be satisfied:$1

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$(-\infty, 4)$

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(A) Given the quadratic equation $f(x) = x^2 - 2kx + k^2 + k - 5 = 0$.For both roots to be less than $5$,the following conditions must be satisfied:$1$. Discriminant $D \ge 0$: $D = (-2k)^2 - 4(1)(k^2 + k - 5) = 4k^2 - 4k^2 - 4k + 20 = 20 - 4k \ge 0 \Rightarrow k \le 5$.$2$. $f(5) > 0$:$f(5) = 5^2 - 2k(5) + k^2 + k - 5 = 25 - 10k + k^2 + k - 5 = k^2 - 9k + 20 > 0$.$(k - 4)(k - 5) > 0 \Rightarrow k \in (-\infty, 4) \cup (5, \infty)$.$3$. Vertex position: $-\frac{b}{2a} $-\frac{-2k}{2(1)} Taking the intersection of all three conditions: $k \le 5$,$k \in (-\infty, 4) \cup (5, \infty)$,and $k < 5$,we get $k \in (-\infty, 4)$.

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:

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