Possible zeros are ± the factors of 60:

±1, ±2, ±3, ±4, ±5, ±6, ±10, ±12, ±15, ±20, ±30, ±60

It's a matter of trial and error: 

try -1

-1| 1 -3 -21  43  60
  |   -1   4  17 -60
    1 -4 -17  60   0

Yes 1 is a zero. So the factorization so far is

(x-1)(x³-4x²-17x+60)

The possible zeros of x³-4x²-17x+60 are still

±1, ±2, ±3, ±4, ±5, ±6, ±10, ±12, ±15, ±20, ±30, ±60

try 1

 1| 1 -4 -17  60
  |    1  -3 -20
    1 -3 -20  40

1 is not a zero of x³-4x²-17x+60

try -1

-1| 1 -4 -17  60
  |   -1   5  12
    1 -5 -12  72

-1 is not a zero of x³-4x²-17x+60
 
try 2

 2| 1 -4 -17  60
  |    2  -4 -42
    1 -2 -21  18

2 is not a zero of x³-4x²-17x+60

try -2

-2| 1 -4 -17  60
  |   -2  12  10
    1 -6  -5  70

-2 is not a zero of x³-4x²-17x+60

try 3

 3| 1 -4 -17  60
  |    3  -3 -60
    1 -1 -20   0

Yes 3 is a zero of x³-4x²-17x+60. So the factorization so far is

(x-1)(x-3)(x²-x-20)

We can factor the trinomial in the last parentheses without
using synthetix division:

(x-1)(x-3)(x-5)(x+4)

That's the final factorization:

The zeros are 1,3,5,-4.

Edwin