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The given expression to factor is:

\[
(x^2 + 9)^2 - 36x^2
\]

This resembles the form of a difference of squares, which is:

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

In this case:

- \( a^2 = (x^2 + 9)^2 \), so \( a = x^2 + 9 \)
- \( b^2 = 36x^2 \), so \( b = 6x \)

Now apply the difference of squares formula:

\[
(x^2 + 9 - 6x)(x^2 + 9 + 6x)
\]

This gives us the factored form:

\[
(x^2 - 6x + 9)(x^2 + 6x + 9)
\]

Notice that each of these factors can be simplified further:

- \( x^2 - 6x + 9 \) is a perfect square trinomial, which factors as \( (x - 3)^2 \)
- \( x^2 + 6x + 9 \) is also a perfect square trinomial, which factors as \( (x + 3)^2 \)

Thus, the fully factored form of the original expression is:

\[
(x - 3)^2(x + 3)^2
\]

This matches option **B**:

\[
(x - 3)^2 \cdot (x + 3)^2
\]

Would you like a deeper explanation or have any questions?

### Five related questions:
1. How can the difference of squares formula be applied to more complex expressions?
2. What other types of identities are useful for factoring polynomials?
3. How would you expand \( (x - 3)^2(x + 3)^2 \) back into its original form?
4. Can the expression \( (x^2 + 9)^2 - 36x^2 \) be factored using another method?
5. Why is it important to recognize perfect square trinomials in factoring?

### Tip:
Always look for special identities like the difference of squares or perfect square trinomials to simplify factoring.

Factor the expression (x^2 + 9)^2 - 36x^2.

Learn how to factor the expression (x^2 + 9)^2 - 36x^2 using the difference of squares formula. This solution explores key algebraic concepts including factoring perfect square trinomials. Suitable for Grades 9-11.

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