If the emphasis is on whose car, or which particular car you went in, then it would be correct to say:
I went to Los Angeles in my/Peter"s/his/a rented car.
If the emphasis is on the fact that you travelled in a car and not by any other form of transport, then Goober"s example is correct:
I went to Los Angeles by car (I did not go by train, or by boat).
It is not normal in English to say: I went to Los Angeles with a car. with is used to express accompanied by as in: I went to Los Angeles with my girlfriend.
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Hmm, to me "I'm here with my car" means that the person speaking is somewhere accompanied by his car as if his car is a person. This should only be used for people not objects. People do say "I came here with my car" but it isn't correct grammatically. They should say "I came here by car". Well, spoken and written english can be so different sometimes and people do say loads of things that are accepted even if they aren't correct grammatically speaking!