A PCIe 16x slot can run at 1x, 4x, 8x, or 16x depending on hardware support and BIOS settings. Unfortunately, in your case, it seems to be a limitation of either the Intel chipset or how HP have configured the whole system - I
think this is your system:
http://www.hp.com/ctg/Manual/c01549447.pdf
If so, then page 75 seems to cover the issue:
Edit: Just noticed another part of that document:
Does your motherboard have two long PCI Express slots? If so, try putting the graphics card into the other slot.
Edit 2: Here's another document (hopefully it's still the right one!)
http://h10032.www1.hp.com/ctg/Manual/c01563108
Here you can see the two PCI Express slots - the white one, although it is a full 16x slot only runs at 4x, whereas the black/brown one runs at full 16x rate.