The sum of all integers x satisfying ((x-2)(x-5))/((x+1)(x-4))≤ 0 is

The critical points are $-1,2,4,5$. For large $|x|$ the expression is positive, and it changes sign at each critical point, so it is negative on $(-1,2)$ and $(4,5)$, positive on $(2,4)$. Zeros of the numerator, $x=2,5$, are allowed (value $0$) but the zeros of the denominator, $x=-1,4$, are excluded. The solution set is $(-1,2]\cup(4,5]$, containing the integers $0,1,2,5$. Their sum is $8$. Wrongly including $x=4$ would give $12$.