If x+3 is a factor it will divide evenly into this polynomial. And the quotient will be the other factor. So we divide to find the other factor.
Synthetic division is easier than long divison for divisors like x+3. You just have to remember that synthetic division is designed for dividing by x-a. So we have to look at x+3 as if it was x-(-3):
-3 | 1 2 -5 -6
---- -3 3 6
----------------
1 -1 -2 0
As the zero remainder (in the lower right corner) tells us, x+3 did divide evenly. And the rest of that bottom row tells us what the other factor is. "1 -1 -2" tells us the other factor is: . So
The second factor is a trinomial that is simple to factor: