synthetic oil versus regular on 2007 Cadillac DTS

I called to set up appointment for an oil change for my 2007 Cadillac DTS and the service tech asked me whether it took synthetic oil or regular. I have no idea which I should use. I have never heard of synthetic. I checked my manual and it doesn't mention synthetic. I also looked under the hood for some kind of sticker but couldn't find one. On the oil cap it just specifies the weight of the oil. I want my car to last me a long time and want to do the right thing for it. Does it really matter and how do I know which oil I should use for it?

Asked by for the 2007 Cadillac DTS
My book shows synthetic oil (5W-30) for cold climates and if you want maximum protection