The board you have, does not have 5.1/7.1 connections on the rear of board itself. It only has three 3.5mm sockets on the board itself, which are stereo out (green), line in (blue), mic (pink/red).
The board does support 8 channel audio (5.1/7.1), but it seems separate 5.1/7.1 connections need to be plugged into the board.
There is additional note on the Asus site relating to the onboard audio. "*2: Use a chassis with HD audio module in the front panel to support an 8-channel audio output."
Can't say I have ever seen a case with front case/panel 5.1/7.1 connections, only stereo out and mic. Most front bay devices that offer audio sockets, are also only stereo out and mic. Never seen one with 5.1/7.1 connections.
I think you are going to have to buy a sound card, even if it's only a cheap one.