A square root is complex if it's applied to a negative number. So we need to find out when the quadratic <0.
-3x2-x+24<0, 3x2+x-24>0, (3x-8)(x+3)>0. This is true if both factors are positive or both factors are negative. So:
Positive: 3x>8, x>8/3, x>-3. Since 8/3>-3 we choose x>8/3;
Negative: 3x<8, x<8/3, x<-3. Since -3<8/3 we choose x<-3. Therefore the square root is complex if x>8/3 or x<-3.