Algebra: How do you factor a trinomial with 2 negative integers? (Ex: s2-4s-12 )

The product of the two last numbers in the factors must be - 12, so one must be + and the other -

that means combinations like
1 -12
12 -1
2 -6
-6 2
4 -3
-3 4
Then they have to differ by 4 to get that - 4 in the middle
that means
6, -2 or 2,-6
so
(x-6)(x+2)