It's entirely possible that your motherboard simply does not support the 3rd gen Intel processors, even with the latest BIOS update (if one exists). Since HP does not provide anything in regards to a BIOS update, it's also entirely possible that attempting to update it from a third party could render your system useless.
Since the
System Specifications does not list the motherboard model number, you will need to take the cover off of the system (after proper shutdown and power disconnect) and look for a model number on the motherboard itself. You may be required to remove any expansion cards in order to see if a model number is printed on the board.
With the motherboard model number, you should be able to locate, on the...