The very most basic type of factoring is removing a common factor:
You started with this:
4m - 8
---------
4 - 2m
Remove a common factor of 4 from the two terms of the numerator, and a common factor of 2 from the two terms of the denominator:
4(m - 2)
----------
2(2 - m)
The numerator and denominator have a common factor of 2, which you can divide out. Now you've got this:
2(m - 2)
---------
(2 - m)
Do you see that (m - 2) and (2 - m) are OPPOSITES? And when you divide a number by its oppposite, you always get -1.
So, you've got
2 * [ (m - 2) / (2 - m)]
or
2*-1
or
-2
If this does not make sense to you, you probably should talk to your teacher.