The expression to factor is \( 16x^4 - 625y^4 \). 

This is a **difference of squares** because it can be rewritten as:

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

The difference of squares formula is:

\[
a^2 - b^2 = (a - b)(a + b)
\]

Using this formula:

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

Next, notice that \( 4x^2 - 25y^2 \) is again a difference of squares:

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

So the fully factored form of the original expression is:

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

This is the complete factorization of \( 16x^4 - 625y^4 \).

Would you like a deeper explanation of any step?

Here are five related questions:
1. How do you factor a sum of squares, if possible?
2. What is the difference between a sum of cubes and a difference of cubes?
3. How can you apply the difference of squares method in real-world problems?
4. What are the conditions under which factoring is possible?
5. How would the factorization change if we introduced higher powers?

**Tip:** Always check if a quadratic or higher degree expression can be factored further using common techniques like grouping or the difference of squares!

Learn how to factor the algebraic expression 16x^4 - 625y^4 step by step using the difference of squares method. This solution covers key algebraic concepts like factoring and the difference of squares formula. Suitable for students in Grades 9-12.

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