BIOSTAR H67MU3 LGA 1155
Radeon HD 6870
700 W Psu
could run modern games?
in short, yes it could.
but, with a core i3 + h67 combo you lose overclockability, aes-ni.
with the motherboard you lose ram upgrade (2 slots already filled by the 2x 4gb dimms, but most games and apps hardly use more than 4 gb), dual gfx cfx/sli, imo the ram slots' placement is a bit weird and might
block upward heat flow. and i hope your gfx card doesn't interfere with the analog jacks and the sata ports. but i have no experience with this motherboard ,so i cannot verify this.
for video editing you should benefit from more cpu cores(4+) and system ram(16+ gb).
for this setup the 700w psu might be a bit overkill.
see if any of these
fit your needs better.
you could specify your current specs and budget in the new build form