In the end this is a matter of Cal. law and I am not a Cal lawyer. But on general principles, you can probably show a picture of the car with no identifying features. But using her name without permission can get you sued. And someone can write a negative review for any reason or no reason. So I would back off in this customer’s case.
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I would take the picture down. If you have a customer who is unhappy because she feels that you are violating her privacy, she is within her rights to write a negative review (which she could do regardless.) A negative review online is a serious thing in terms of your search engine optimization.
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While I know of no legal reason why you cannot publish a photograph of a former customer's car in an advertisement for your business, as my colleagues note you most assuredly do not want the negative review. So remove that photograph pronto, apologize to the former customer, and let her know you meant no insult or harm. And tell her you will wash her car for free next time. The investment in this kind of customer relations will come back to you many fold when, by word of mouth, she tells others how considerate you are.
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Legally, your rights are to post the photograph of the car to showcase your work, provided you don't falsely imply an endorsement, either from the car company or the customer.
You are, however concerned with the wrong person's rights. Your customer has the right to post negative reviews and cost you business. You have your priorities backwards, and should worry about your customer more and your legal rights less if you want to succeed in business.
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