IPHONE 4 SLOW

What would be the best solution to improving the speed in my chicks iPhone 4?  Obviously upgrading to the 4s or 5 will solve that issue but that isnt an option at this point.  Ive tried getting her to get a new iphone but she doesnt want to... She complains about the phone being slow on certain apps. 

With that said, would doing a factory reset, and then restoring one of her backups resolve the issue?  or is this just something she's going to have to deal with until I finally just go out and buy her a new phone.

Thanks

By the way, the phone is fully updated, and all the background apps have been X'd out of.