Assume for the moment that "the car" (automobile) was invented at an identifiable time. The word "before" in the test question renders the past perfect tense ("had been invented") unnecessary and verbose.
The short answer is that either "was invented" or "had been invented" is good enough, but "was invented" is possibly better, because of its brevity.
Also for brevity, I would strike "Back in the days," saying, "Before the car was invented, the only personal means of transportation were the horse and the bicycle."
An example of a sentence in which the past perfect is necessary is, "When roller skates were first produced, the automobile had not been invented."
- The word 'car' dates from around 1300. It means "wheeled vehicle." Before the automobile, there were other modes of personal transportation besides horses and bicycles. For example, roller skates were invented in the mid 1700's.