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I think this depends on what made the other relationships bad. Was it a behavior that one of you was doing? Was it just that you two weren't mutually compatible?

If it's a behavior I'd say that it's important to work on whatever that thing is first. If it's just that you're not compatible with each other, remember that not every relationship is going to be perfect and not every person is going to be "the one." You have to go through the people who are wrong for you before you land on the person who is right. If it's just a compatibility issue, it might be worth giving things a try since you really like this girl.

Communication is important in any relationship, so if you do end up trying things out with this girl make sure to talk to her. You can let her know if you are trying to improve yourself and want the most out of this relationship. Keep the lines of communication open throughout the relationship. Even if you end up not going out with her, you can just tell her that you are flattered but right now you're not looking for a relationship.