Doesn't sound right. I'd use just "had"

Your so-called correct answer is wrong. The phrase 'before it broke down' fixes this situation in the past. It is a finished or 'closed' period of time, so you cannot use a present perfect here. There is no connection with present time, so need a past tense verb - either past simple or past perfect. I should be: We had a car for ten years before it broke down. It's also possible to use a past perfect 'We had had..' in some situations.

The word HAVE is doing double duty in this situation. So to understand it easier substitute "owned" for had. Then: We had owned a car for ten years before it broke down. This seems correct and implies that the car is not owned anymore because it is broke. Therefore, the answer according to grammar rules is "had had" However, 90 % of native speakers would not say "had had", but rather would just drop one "had". Saying the same word twice in a row sounds weird.

We had a car for ten years before it broke down. We had had a car for ten years before it broke down. But, stylistically, I always use simple past "had". "Had had" is weird. "Have" makes no sense in your example. It implies that the statement is true today, which in your example it is not. We have had this car for ten years. (and we still have it today). We have always had Chevys in my family. (everyone is my family has had a Chevy in the past, and we all have Chevys now). I think to say, "we have had this car" implies that you can use it, which is why to put it next to "before it broke down" doesn't work well in the sentence. Compare: My dad has had these broken cars parked on the lawn for two years. (they were broken in the past, and my dad had them then, and they're still broken today and my dad has them)

Past perfect continuous: had (past tense) had (perfect) ____ing (continuous/progressive) Therefore: We had had the/a car for ten years before it broke down. When you're talking about a specific object, it's better to use 'the'. 'a' is an indefinite article - it can mean any car. So 'the' means you're talking about the car that had broken down.