Inspiron 15 5566 - Dual Monitor Support?

Unfortunately no, HDMI has no native support for multiple independent displays driven by a single output; the splitters you'll see on the market would only mirror the same source image to multiple destination displays.  Only DisplayPort (including DisplayPort running over USB-C) allows multiple displays to be driven off of a single output.  For multiple external displays, your only option would be to use a USB adapter that contains a DisplayLink chip, but those do not work the same way as displays that are natively driven by the GPU.  DisplayLink works by using the CPU and GPU to compress image data before transmitting it as raw USB data to the adapter (unlike USB-C, which has an actual GPU output wired to it), and the adapter then decompresses that USB traffic and outputs the result to the display.  However, this can have some adverse consequences:

- When large areas of the display are changing simultaneously, e.g. when gaming or watching full screen video, you may see compression artifacts, or the "fluidity" of the display may vary. For example, full screen video might alternate between bogging down and then "fast-forwarding" to catch back up, so it averages out to being in sync with the audio, but it's not a great experience.

- The above behavior can also occur when other devices/tasks are consuming significant USB bandwidth (such as a large transfer to/from an external hard drive) or your CPU is heavily utilized by other tasks and therefore doesn't have as much capacity to deal with this display compression work.

- The extra CPU and GPU utilization will noticeably decrease battery life if you use the external display while on battery power.

Bottom Line: If you just need an extra display for things like email and other basic productivity tasks, then DisplayLink adapters can be a perfectly acceptable solution, since after all it's better from a productivity standpoint than not having an additional display at all.  But if you want to use it for gaming and similar types of tasks, you probably won't have a great experience.