I can't connect my Solus 3 Budgie laptop with a projector and doesn't work with HDMI neither, it does not seem to recognize them. I tried with Fn+F4 keys combination and through Display settings, the external display (projector and TV) did not appear.

Whit Thomas Lucio and @STiAT guidance, on G+ community, I ran some process (results are in attached file).

At saying of Georg "is certainly possibly we're missing some intel bits there (firmware / kernel config, and who'd better know than Ikey".

The laptop it's HP 240 G4, relevant especs are:
CPU: Intel Celeron N3050 (1,60 GHZ hasta 2,16 GHz Turbo) 1600MHz/2 MB level 2, dual-core, 6 W
Integrated graphic card: Intel HD
Dedicated graphic card: AMD Radeon R5 M330 (DDR3 2 GB dedicated, interchangeable)
RAM: SDRAM DDR3L - 1600 2048 MB (2048 MB x 1)
VGA and HDMI outputs.

Solus "Details" it shows as:
Processor: Intel® Celeron(R) CPU N3050 @ 1.60GHz × 2
Graphics: Gallium 0.4 on llvmpipe (LLVM 4.0, 128 bits)

Thanks in advance.