
Lets check with every option
(a) [-4,-3]
We plug in -4  for x  and -3 for x


f(-4) is positive and f(-3) is negative. there is some value at x=c on the interval [-4,-3] where f(c)=0. so there exists atleast one zero on this interval.
(b) [-3,-2]
We plug in -3  for x  and -2 for x


f(-2) is negative and f(-3) is negative. there is no value at x=c on the interval [-3,-2] where f(c)=0.  
(c) [-2,-1]
We plug in -2  for x  and -1 for x


f(-2) is negative and f(-1) is positive. there is some value at x=c on the interval [-2,-1] where f(c)=0. so there exists atleast one zero on this interval.
(d) [-1,0]
We plug in -1  for x  and 0 for x


f(-1) is positive and f(0) is negative. there is some value at x=c on the interval [-1,0] where f(c)=0. so there exists atleast one zero on this interval.
(e) [0,1]
We plug in 0  for x  and 1 for x


f(0) is negative and f(1) is negative. there is no value at x=c on the interval [0,1] where f(c)=0.  
(f) [1,2]
We plug in 1  for x  and 2 for x


f(-4) is positive and f(-3) is negative. there is some value at x=c on the interval [-4,-3] where f(c)=0. so there exists atleast one zero on this interval.
so answers are (a) [-4,-3], (c) [-2,-1],  (d) [-1,0], (f) [1,2]