do you have a video card with hdmi port?
The onboard audio has been disabled in the BIOS.
No HDMI video card- the card is a PCI Audigy 2 card.
In the screenshot, I see HDA NVidia, HDMI 0 (HDMI Audio Output), which appears to be the hub. Then, there are 4 ports, HDA NVidia, (HDMI 0) #3, #7, #8 and #9. So, you have HDMI audio coming from somewhere
I have no experience with HDMI, so I'll leave the floor to T6.