Rewrite $ 64x^2y^6-1 $ as:
$$ \color{blue}{ 64x^2y^6-1 = (8xy^3)^2 - (1)^2 } $$Now we can apply the difference of squares formula.
$$ I^2 - II^2 = (I - II)(I + II) $$After putting $ I = 8xy^3 $ and $ II = 1 $ , we have:
$$ 64x^2y^6-1 = (8xy^3)^2 - (1)^2 = ( 8xy^3-1 ) ( 8xy^3+1 ) $$