To my mind, it looks at a very difficult stage to root. Usually roses are either rooted as hardwood cuttings, when the rose is going dormant, or semi-hardwood cuttings which are about the same as a deadhead. That cutting is going to try and put all its energy into growing leaves, and trying to get it to switch gears and grow roots is going to be difficult/impossible.

You can give it a try as you have nothing to lose. There are many ways to do it and a quick search here will come up with a lot of them. I use seed starting soil and rooting hormone for mine.

Hi Tori. Like mentioned above, there's nothing to lose by trying it, but if it doesn't work don't be too disheartened.

If you can get a cutting of the same rose that is more mature it may work better. Try using a cutting that is thicker than a pencil and doesn't have that new red growth, but instead has a spent bloom on it (cut off the spent bloom). You may have better luck with that. But you can try this one to see what happens. :)

Thank you all for the reccomendations. Sadly I got the little sprig while helping someone move while I was with my grandparents. I don't think I will be heading back there anytime soon so this is my one shot at this mystery rose.