1. Do not use FOIL, because the problem is better solved as it is.
2. Visualize your numbers on the number line.
3. Only one of the two conditions on each side of (OR) is viable.
(x+3 <0 and x-5>0) OR (x+3>0 and x-5<0)
(x<-3 and x > 5) OR (x> and x<5)
(x<-3 and x>5) OR (-3<x<5)
It is impossible to have x<-3 AND x>5, so this case is eliminated. So (-3<x<5) is the right answer.
This question has been answered three times before, you can go through the different answers.