Look for rational zeroes first. The magnitude of these will be factors of 10: 1, 2, 5, 10.
We can have positive or negative values of these so let's start with 1 and -1. Even powers of x will give us positive coefficient values: 1-5+7+3-10=-10 so 1 is not a zero (root); 1+5+7-3-10=0 so -1 is a zero.
Using synthetic division we can divide by the zero:
-1|1 -5 7 3 -10
1 -1 6 -13 | 10
1 -6 13 -10 | 0 = x3-6x2+13x-10.
Now we look for zeroes of the cubic. Neither 1 nor -1 are zeroes, so try 2:
8-24+26-10=0, so 2 is another zero. Divide by it:
2|1 -6 13 -10
1 2 -8 | 10
1 -4 5 | 0 = x2-4x+5 has no real zeroes (it has complex zeroes).
We can find the complex zeroes:
x2-4x=-5,
x2-4x+4=-5+4=-1,
(x-2)2=-1,
x-2=±i,
x=2±i.
We now have all 4 zeroes: -1, 2, 2+i, 2-i.