There is nothing in the construction of this sentence that would imply a plural version of "dommage". In my mind, a car might have several countable instances of damage (which would imply "dommages") but one general state of damage (so I would use "dommage"). Clearly this is wrong thinking on my part. Is there a good rule when to use "dommage" vs. "dommages"?
To phrase the question differently: Why souldn't I translate "a lot of damage" as "beaucoup de dommage"?
EDIT : My previous answer was confusing and inaccurate. I'll just replace it by what I found after some research :
The English word "damage" can be both countable and uncountable.
The French word "dommage" can be both countable and uncountable.
Therefore, "beaucoup de dommage" should be accepted.
I think you have to apply some French logic. In English, damage in this context is always singular, no matter how many dents and scrapes you can count. In French, if there is more than one spot of damage, it's plural. So beaucoup tells you there must be a lot of damage, which in French can not be singular. Therefore it has to be dommages not dommage, even though they sound the same.
Jame-holden2 thanks for your comment but I still hold out that in 25 years of living in France and in ten years teaching English on a course for insurance law and insurance in the university of Poitiers I never saw "dommages" in the plural used for damage in the sense of physical damage to a car or anything else. I only ever saw the plural form "dommages" being used to describe "damages and interests" as in compensation awarded by a court.
Do French speakers say "...a beaucoup de dommages." or "...a beaucoup de dommage." or both? I have found more examples of "...a beaucoup de dommage." online than "...a beaucoup de dommages.", with the main entries of the latter being Duolingo! If this usage is not normally used by French speakers then this example should be removed and replaced with the more commonly used construct.
In American English one could say either damage or damages with respect to this car.
Looking the car over you say to the seller, "the headlight is broken, the fender is bent, the taillights are damaged, the line for the oil is clogged an the gas tank leaks now.
Wow! This car is damaged here, there and everywhere. It has a lot of damages!" or "the car has a lot of damage." We could say this either way and be correct.
So damage and damages should both be accepted. 8/30/2018