As far as I can tell, if you want to say something like "Would you...?" or "Why don't you..?", it's common to say something like "食べるのだろうか？". I looked it up and as I far as I can tell it's short form+の to make it a noun, and then だろう which is the volitional of だ... this is after a lot of Googling around and I'm still kind of confused, since I haven't even heard the volitional mentioned in my class yet. Please tell me if that's right.
If I understand Wikipedia correctly, you could also say "食べようか？". First off, is that conjugating it right; second, is that something you would say; finally, how is it different from -の+だろう?
Thanks for any help.