<human>: Can you give me the gist of the text in a nutshell?

Alright, so I've been talking to this girl for about six months now and dating her for four. I'm seriously falling for her. 

She broke up with her ex about one and a half years ago. She dated him for four or five years, moved to another country with him, then moved back here and broke up with him. When we first met she told me that they were still friends, which I was fine with, and that they still told each other they loved each other, which was a bit of a red flag for me, but I didn't know how it would go with this girl, so I didn't raise a stink. Some friends tell each other they love each other, so whatever, right?

Cut to now, and she's facebook messaging him a lot. She went on her computer to show me something a few weeks ago, and she had her facebook inbox up, which she quickly closed out of, but not before I saw the last message she had sent her ex, which was "Love you baby <3". Now, the love you stung, but it was the inclusion of baby that really bothered me.

Again, last night, same scenario, this time the last message from him read "I blame it on your perfect body."

I really want something with this girl and I've really fallen for her in the time I've known her. She's distant at times, but treats me really well. We have a ton in common, have great sex, and it doesn't hurt that she's all around adorable. But I can't have something with her if she still has feelings for her ex. It's been eating at me since last night.

I'm unsure how to approach this. I don't think I was snooping, but I'm afraid it will sound like I was. How should I go about it?
<bot>: Read a couple flirty messages ("Love you babe <3" and "I blame it on your perfect body") between GF and her ex. It's eating at me and I need to know how to approach her about this.