There's a straightforward albeit messy way to express a new polynomial with those roots in terms of just a,b,c. The function I wrote above in terms of alpha and beta is symmetric in alpha and beta. Without loss of generality, let and let . Substitute in and you have your equation. (Choice of is arbitrary; for simplicity, you can let .)
I still don't know what setting is supposed to accomplish, but maybe I could figure it out by looking at an example in your book. Possibly it's something obvious that I'm just not seeing. If you have the intended solution and wish to post it, I'd be curious to see.