Why will iTunes no longer sync?

So a couple of days ago I restored my laptop to factory settings, re-installed itunes and attempted to add new songs to my iPhone like I did before the restore.iTunes wants to erase the music on my phone just to add 3 new songs. I click manually manage songs and then press sync and nothing new is synced. I just want all of my music back on my iTunes and to be able to plug in my iPhone and it automatically sync new songs!

Because you restored your computer to factory settings, your computer was not recognized with the iPhone anymore. Therefore, you lost all your songs.

When the iPhone has to sync with a computer for a first time, when you put songs it has to sync the iPod music library to match the computer's iTunes library. That's why you only have 3 songs, because on your iTunes (computer) you only had those 3 songs. Technically speaking, if you want all your songs back, you need to get them back. Even before you synced it, you were doomed of that happening.

Bottom line, you need to get all your music back to put it onto your iPhone there's no other way.
Sorry bud.

EDIT NOTE: Diana B. I understand what you're trying to say but the computer is no longer authorized with the IDevice because the computer was restored to factory settings (meaning putting it back to when it was bought). That means it would have to sync once at least to be able to even work on the computer, there's no avoiding what happened. To add a song, it needs to delete all music library and update it with the current iTunes library on the computer. And plus it's already happened so there's no avoiding that anymore.

To the one who asked the question, I'm sorry but you'll have to get all your music back by scratch. But what I do suggest is to save all your music and make a copy of it on your computer, keep it on a USB or back up your iPhone with all the current music library so if you restore your iPhone from a back-up it'll boot up with all your songs on it again.