Do you have a 64 bit win7 Pro? You would need win7 Pro or Ultimate 64 bit for 16 GB.
Usually, you need to slot in matched pairs of modules in your 4 slots. I believe your system has a maximum memory of 16 GB. A bank is considered as 2 slots. A 4 slot would mean 2 x 2 slots. This essentially implies that you need matched pairs.
Check if you got any installation guides with the memory kit or try talking to the supplier. Advise them their sticks does not work.
Before you do this, try the following:
Keep the original stick on socket 1 and put a new one on 3. See if that works. If it did, put the rest on 2 and 4. I am not sure if this would work as they are not matched pairs.
If all fails, you should be able to return them and buy on line with a seller who would sell you matched pairs.
I do not know anything about this seller. Try them, if you like.
http://www.memorystock.com/ram/HewlettPackardPavilionp6823wRAM.html
If you have to return and buy a new one, I am afraid that you are going to have remove the HP stick and buy two matched pairs of 2 x 8 (2x4GB).
Let's us know how you got on...good luck