socal - sorry, absolutely not. A 32-bit system cannot possibly runa 64-bit version of anything. You will, however be able to upgrade to 32-bit Win 7 and will notice no difference. You will be restricted to a maximum of about 3.2 Gb of memory, but in actual fact, Win 7 runs better than Vista with the same amount of memory.
You could also check the HP website in case your PC really has a 64-bit processor and motherboard, but simply has a 32-bit version of Windows installed. Most AMD processor over the last several years have all been 64-bit, but few have been running 64-bit, mostly because of poor support for hardware drivers on 64-bit platforms. That problem is now much diminished.
For example, my PC is over 4 years old, came with a 64-bit AMD processor, but 32-bit Windows XP-home. I upgraded the memory to 2Gb and XP-pro. Still 32-bit. Lately, I dual-boot a 64-bit Win 7 release candidate which runs perfectly and supplies all the necessary 64-bit drivers for my existing hardware by itself.
Good luck and dont worry about it.....