To factor the expression \(125x^4 - 216xy^6\), we can follow these steps:

1. **Identify common factors**:
   The greatest common factor (GCF) of \(125x^4\) and \(216xy^6\) is \(x\). So, factor out \(x\) first:
   \[
   x(125x^3 - 216y^6)
   \]

2. **Recognize a difference of cubes**:
   The expression inside the parentheses, \(125x^3 - 216y^6\), is a difference of cubes:
   \[
   125x^3 = (5x)^3 \quad \text{and} \quad 216y^6 = (6y^2)^3
   \]
   So, we can apply the difference of cubes formula:
   \[
   a^3 - b^3 = (a - b)(a^2 + ab + b^2)
   \]
   Here, \(a = 5x\) and \(b = 6y^2\).

3. **Apply the difference of cubes formula**:
   \[
   125x^3 - 216y^6 = (5x - 6y^2)((5x)^2 + (5x)(6y^2) + (6y^2)^2)
   \]
   Simplify each term:
   \[
   (5x)^2 = 25x^2, \quad (5x)(6y^2) = 30xy^2, \quad (6y^2)^2 = 36y^4
   \]
   So the factorization becomes:
   \[
   125x^3 - 216y^6 = (5x - 6y^2)(25x^2 + 30xy^2 + 36y^4)
   \]

4. **Final factored form**:
   Substituting this back into the expression, the complete factorization is:
   \[
   x(5x - 6y^2)(25x^2 + 30xy^2 + 36y^4)
   \]

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Would you like more details on any part of this factorization?

Here are 5 related questions you might find useful:

1. How do you factor a sum of cubes?
2. Can the quadratic term in the factorization be factored further?
3. What are the applications of the difference of cubes formula?
4. How do you factor trinomials with three variables?
5. How do you determine the greatest common factor (GCF) of polynomial terms?

**Tip**: When factoring, always look for common factors first before applying more advanced techniques like special binomial identities.

This guide explains how to factor the expression 125x^4 - 216xy^6 using the difference of cubes formula and GCF method. Suitable for high school students (Grades 10-12), the step-by-step solution covers advanced algebraic concepts.

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