germanic barbarians invaded the roman empire in the 4th A.D. because
they were being pushed westward by
1. the carolingians
2.the Huns
3.the Hittites
4.the Bulgars

Hello.

The answer is 2. The Huns.

sources: 

"The Goths, a Germanic people, erupted into the region from the Baltic
area about A.D. 200 but were pushed westward by the Huns in A.D. 370.
"
source: scholastic.com
http://teacher.scholastic.com/scholasticnews/indepth/cold_war/history_rus.asp

"The Goths, ... Germanic people, had never been really conquered by
the Romans. They had through centuries of contact become partly
Romanized. They provided the best soldiers to the Roman armies. They
were fierce, larger men, and independent in spirit. They were being
slowly driven westward by the Huns coming from the east."
source: Roman civilization, hosted by sympatico.ca
http://www3.sympatico.ca/nimbus/z23f.htm

search strategy:
"westward by the huns"

I hope this helps.