There is no specific rule to factorisation.
I personally have always started with the smallest number I can think will divide the number (apart from one).
I would have attempted 120 this way.
120
/ \
2 x 60
/ \
2 x 30
/ \
2 x 15
/ \
3 x 5
As you can see above, I've continually divided 120 by the 2. Therfore, the factorisation of 120 is 2 x 2 x 2 x 3 x 5. Done!
Supposing I divided by 3 first instead.
120
/ \
3 x 40
/ \
2 x 20
/ \
2 x 10
/ \
2 x 5
The factors I get this time are 120= 3 x 2 x 2 x 2 x 5.
As you can see, I doesn't matter which number I factorise out of the original number first. In the end you get the same answer.
Supposing I divided by 4 first as well.
120
/ \
4 x 30
/ \ / \
2 x 2 3 x 10
/ \
2 x 5
The factors I get this time are 120= 2 x 2 x 3 x 2 x 5.
As you can see, if you divide by four, you have to do an extra step as four can been divided by 2 as well.
You should always start with the smallest number you can think of which can completely divide the original number (apart from 1!).
The teacher's method was original correct as she started with 2. You started with 4 which could be divided further as well. As you as you did what I did with the "division by 4" above, you should be fine.
The teacher should accept both of your methods in fact.
36 = 2 x 2 x 3 x 3 is as correct as 36 = 3 x 2 x 3 x 2
Don't worry! Maths is very flexible! 6 = 3 x 2 is the same as 6 = 2 x 3, right?!
Any other problems, feel free to email me at atif.muhammad@gmail.com