We memorize the way to factor the sum or the difference of cubes:

A³ ± B³ = (A ± B)(A² ∓ AB + B²)

That is, the sum of two cubes factors like this:

A³ + B³ = (A + B)(A² - AB + B²)

And the difference of two cubes factors like this:

A³ - B³ = (A - B)(A² + AB + B²)

Be sure to memorize these. 

Your problem is

64x³ - 27y³

since 64 = 4³ and 27 = 3³, we can write that as

(4x)³ - (3y)³

So we substitute A = 4x  and B = 3y  in

A³ - B³ = (A - B)[A² + AB + B²]

(4x)³ - (3y)³ = (4x - 3y)[(4x)² + (4x)(3y) + (3y)²]

Then we simplify the bracket expression:

              = (4x - 3y)[16x² + 12xy + 9y²]
 
              = (4x - 3y)(16x² + 12xy + 9y²)

Edwin