How to delete song from iTunes forever?

I deleted songs off of my iPhone but when I updated my phone to the new software, every song and movie I've purchased from years back is now visible. I want to delete them not only from my iTunes and iPhone, but entirely so that when another update occurs, these songs don't even show up ever again. Is this possible and if so, how do I go about doing this?

I don't think you can if you paid for it.