HDMI doesn’t work with Optimus hybrid graphics chipsets under Linux for most laptops, even if you use Bumblebee, but there is a way to get HDMI to work: by using a separate session.
Using the instructions below, you’ll get an LXDE session (you can use XFCE or others, but the instructions below are for LXDE) to show up on the external monitor (which is connected via HDMI), and in this session, all the applications you launch use your laptop’s Nvidia graphics card and you can play games, watch movies, etc. The LXDE session runs in the same time as your regular session, so on your laptop’s screen you get your regular (main) session. Also, the mouse and keyboard are shared between the two sessions / monitors.