The rational zeroes are factors of the constant 6. The list consists of all these factors in their positive and negative forms: 1, -1, 2, -2, 3, -3, 6, -6. The actual (integer) zeroes, if any, are found by plugging in each of these so that f(x)=0.
1: 1+10+14-6=19
-1: -1+10-14-6=-11
2: 8+40+28-6=70
-2: -8+40-28-6=-2
3: 27+90+42-6=153
-3: -27+90-42-6=15
6: 216+360+84-6=654
-6: -216+360-84-6=54
From this list, x=-2 is closest to an actual zero. None of the rational zeroes are actual zeroes.