64x³ + 125y³

There are two ways to do this.

The cube root of 64x³ is 4x and the cube root of 125y³ is 5y

You can use long division to divide by the sum (or difference if it
had been the difference of two cubes) of the two cube roots, like this, 

1. Divide by the sum of the cube roots of the terms
   inserting two placeholder zero middle terms 0x²y + 0xy²

        16x² - 20xy  +  25y²                              
4x + 5y)64x³ +  0x²y +   0xy² + 125y³
        64x³ + 80x²y
              -80x²y +   0xy²
              -80x²y - 100xy²
                       100xy² + 125y³
                       100xy² + 125y³ 
                                   0

And so the factorization is 

16x³ + 25y³ = (4x + 5y)(64x² - 20xy + 25y²)

Or you can do as most people do -- memorize how the sum or
difference of two cubes factors:

A³ ± B³ = (A ± B)(A² ∓ AB + B²)

The double sign symbols ± and the ∓ mean either use the signs 
on top or the signs on the bottom

A will represent the cube root of the first term and B the cube root
of the second term.

Your problem

64x³ + 125y³ 

The cube root of 64x³ is 4x and the cube root of 125y³ is 5y.

So we substitute 4x for A and 5y for B and ± means + and ∓ means -
     
So A³ ± B³ = (A ± B)(A² ∓ AB + B²) becomes:

   (4x)³ + (5y)³ = (4x + 5y)[(4x)² - (4x)(5y) + (5y)²]

     16x³ + 25y³ = (4x + 5y)(64x² - 20xy + 25y²)

Edwin