Sometimes the serial number is only printed on the product packaging. If you still have it, that may be the place to look. Since you bought it so recently it's possible you could just take it back to the store rather than messing with HP.
There is the crutch of it... no packaging and can't find the original receipt. I'm gonig to try going directly through the Houston office on Monday.
Without a receipt you may still get store credit - especially if you make yourself a nuisance to the returns department at the store. Having worked at one I can tell you: no whiner goes unrewarded. This works best at big chains, when there is always someone higher to go to. Eventually you reach someone who will get reamed for bothering their boss about you. If calling Houston dosen't work out this may be a tactic. Let us know how it turns out
tjmutlow, HP claims xcopy causes problems with the drive. Whether this is true you'll have to judge for yourself.
Xenokai, here is the address of Hp's product page for the computer: http://h10025.www1.hp.com/ewfrf/wc/prodinfoCategory?dlc=en&lc=en&product=435559&lang=en&cc=us& I couldn't find the DVD writer model number listed, but since it has 8x write-once speed I don't think it is the hated 300c. Six hundred seems like a lot for the computer you are looking at! Dell.com has had some good deals lately. So has HP.com, if anyone can still trust them
Xenokai, The drive in that system is a LiteOn SOHW-8225 8X DL So you shouldn't have to worry Burners Nec 2500A Pioneer DVR 105
tobingray, You sound like you have the same problem I had. My computer was made to order so when my 8301/44 Philips (which was an OEM HP purchase) didn't work both HP and Philips told me there is no S/N because it was OEM and your dealer had to warrant it. (I learned from a service depot that Philips products with a '/44' are not supported by either Philips or HP in Canada.) I just bought an LG 16X writer for $70.00 with s/w and support and wrote the HP off to bad experience. Never again will I buy HP.