Step-by-step explanation:

a) \(256x^4 - y^4\)

This is a difference of squares. You can factor it as follows:

\[256x^4 - y^4 = (16x^2 + y^2)(16x^2 - y^2)\]

Now, \(16x^2 - y^2\) is also a difference of squares:

\[16x^2 - y^2 = (4x + y)(4x - y)\]

So the complete factorization is:

\[256x^4 - y^4 = (16x^2 + y^2)(4x + y)(4x - y)\]

b) \(3x^2 - 108y^2\)

You can factor out the greatest common factor, which is 3, and then notice that both terms are perfect squares:

\[3x^2 - 108y^2 = 3(x^2 - 36y^2)\]

Now, you can factor \(x^2 - 36y^2\) as a difference of squares:

\[x^2 - 36y^2 = (x + 6y)(x - 6y)\]

So the complete factorization is:

\[3x^2 - 108y^2 = 3(x + 6y)(x - 6y)\]

c) \(4x^2 - 100y^2\)

This is also a difference of squares, with 4x^2 being a perfect square:

\[4x^2 - 100y^2 = (2x + 10y)(2x - 10y)\]

You can simplify this further by factoring out a 2:

\[4x^2 - 100y^2 = 2(2x + 10y)(2x - 10y)\]

d) \(16x^4 - 625y^4\)

This is a difference of squares with both terms being perfect squares:

\[16x^4 - 625y^4 = (4x^2 + 25y^2)(4x^2 - 25y^2)\]

Now, \(4x^2 - 25y^2\) is also a difference of squares:

\[4x^2 - 25y^2 = (2x + 5y)(2x - 5y)\]

So the complete factorization is:

\[16x^4 - 625y^4 = (4x^2 + 25y^2)(2x + 5y)(2x - 5y)\]