Consider x^2-2x-8 = (x-4)(x+2). This is > 0 if either x-4 and x+2 are both > 0 or both < 0. The first case leads to x>4 and x>-2, which is just x>4; the second to x<4 and x<-2, which is x<-2. The reversal of sign comes from this second case where you consider the possibility that two negative factors multiply to give a positive value.