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As long as you're making the minimum monthly payment, you won't have a late payment reported.
Care Credit is the devil though, so if you can swing getting a different 0% card I think it would be the better option for a balance the size of the one you are looking at.
Will the provider allow you to make payments against it?
Also, unless you think you would be approved for a card that's at least 2x(maybe 3x) that, you'll take a UTI hit by putting it on there.
Have you considered some kind of installment loan for it?
Would've gotten back to this sooner but my account was locked for some reason. I have no idea why I asked this. I never carry over a balance so I was intially confused with the relation of APR's and late payments. I ended up apping for the amex blue cash after posting this thread. Never got a large dental bill before so I thought they wanted me to pay upfront and that the work would be all done in a span of a week or so, but apparently it's pay by appointment and after my first visit which was $418 they scheduled a second all the way in Decemeber. Not sure why the manager even tried to sell me on care credit in this case.