hello from mrbilldoesmath!
answer:
(7 x + 6 y) (49 x^2 - 42 x y + 36 y^2)
discussion:
the trick is to recognize that 343 = 7^3 and 216 = 6^3 and, in general,
a^3 + b^3 = (a + b) (a^2 - a b + b^2).
in our case
343x^3+216y^3 = (7x)^3 + (6y)^3
so take a = 7x and b = 6y in the above equation to get the answer.
you,
mrb