(1) x²-x-6<-3x+2,
x²+2x-8<0,
(x+4)(x-2)<0,
Either x+4<0 and x-2>0, implying x<-4 and x>2 (impossible); or:
x+4>0 and x-2<0, implying x>-4 and x<2, so the solution is:
-4<x<2 or x in (-4,2) interval.
(2)
I guess you mean all integers satisfying x²+2x-8<0.
Answer is -3-2-1+1=-5.