2014/054) show that 27x^6+27x^3y^3+8y^6 is composite

We see that

27x^6 + 27x^3 y^3 + 8y^ 6
= 27x^6 - 27 x^3 y^3 + 8y^ 6 + 54 x^3y^3
= (3x^2)^3 + (-3xy)^3 + (2y^2)^3 – 3(3x^2)(-3xy)(2y^2)
Above is
a^3+ b^3 + c^3 – 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - cb) where a = 3x^2, b = - 3xy, c = 2y^2

We are not finished yet. Because one term is –ve we need to show that neither a+b+c is 1 nor other term is 1

a+b+ c = 3x(x-y) + 2y^2 >= 2

a^2 + b^2 + c^2 – ab –bc – ca >=   ½(a-b)^2

or >=  1/2(3x^2+ 3xy)^2 so > 2

as both factors are > 1 so this is composite