hello from mrbilldoesmath!
answer: (7x + 6y) ( 49x^2 - 42xy + 36y^2)
discussion:
in general, the factorization of a^3 + b^3 is
a^3 + b^ 3 = (a + b) ( a^2 - ab + b^2)
in our case note that 343 = 7^3 and 216 = 6^3 so the equation becomes:
343x^3 + 216y^3 = (7x)^3 + (6y)^3
if you think of a (above) as 7x and b as 6y the factorization becomes
343x^3 + 216y^3 =
((7x + 6y) ( (7x)^2 - 7x*6y + (6y)^2) =
(7x + 6y) ( 49x^2 - 42xy + 36y^2)
you,
mrb
(a + b) (a^2 - a b + b^2)