Can an R7 M340 (2GB) with an Intel i7 6500U and 8GB RAM support three monitors? This GPU has only one HDMI, but I was thinking to buy an adapter that is USB 3.0 to HDMI.
"I was thinking to buy an adapter that is USB 3.0 to HDMI" -I wouldn't advise you to do that.
It is not question of R7 M340 capabilities. Contact your laptop manufacturer first of all.They are responsible for functional capabilities of your laptop.
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