To factor the expression \((x^2 + 2xy + 6y^2)(x^2 - 2xy + 6y^2)\), we can treat it as the product of two quadratic binomials. Let's multiply the terms using the distributive property (FOIL method).

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

1. First, multiply the first terms: 
   \[
   (x^2)(x^2) = x^4
   \]

2. Now, multiply the outer terms:
   \[
   (x^2)(-2xy) = -2x^3y
   \]

3. Multiply the inner terms:
   \[
   (2xy)(x^2) = 2x^3y
   \]

4. Multiply the last terms:
   \[
   (2xy)(-2xy) = -4x^2y^2
   \]

5. Now, multiply the cross-terms:
   \[
   (x^2)(6y^2) = 6x^2y^2
   \]
   \[
   (6y^2)(x^2) = 6x^2y^2
   \]
   \[
   (6y^2)(-2xy) = -12xy^3
   \]

6. Finally, multiply the last terms:
   \[
   (6y^2)(6y^2) = 36y^4
   \]

Now, let's combine the like terms:

\[
x^4 + (-2x^3y + 2x^3y) + (-4x^2y^2 + 6x^2y^2 + 6x^2y^2) - 12xy^3 + 36y^4
\]

Simplify the expression:

\[
x^4 + (0) + 8x^2y^2 - 12xy^3 + 36y^4
\]

So, the fully expanded form of the expression is:

\[
x^4 + 8x^2y^2 - 12xy^3 + 36y^4
\]

This is the expanded version of \((x^2 + 2xy + 6y^2)(x^2 - 2xy + 6y^2)\).

Would you like more details or have any questions?

### Related Questions:
1. How do you factor polynomials involving complex roots?
2. What are the properties of quadratic binomials?
3. How do you apply the FOIL method to higher-order polynomials?
4. Can you explain how to recognize special products like sum and difference of cubes?
5. What is the significance of perfect square trinomials?

### Tip:
Always check for common factors or special product patterns before multiplying out polynomials. This can simplify the process significantly!

Factor the expression (x^2 + 2xy + 6y^2)(x^2 - 2xy + 6y^2).

Understand how to factor the expression (x^2 + 2xy + 6y^2)(x^2 - 2xy + 6y^2) using the FOIL method. This problem involves key algebraic concepts like polynomials, the distributive property, and difference of squares. Ideal for students in Grades 9-12.

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