To factor the expression \( a^2b^2d - 9c^2d \), let's follow these steps:

1. **Identify common factors:**
   The common factor between the two terms is \( d \), so we factor out \( d \):
   \[
   d(a^2b^2 - 9c^2)
   \]

2. **Recognize the difference of squares:**
   The expression inside the parentheses, \( a^2b^2 - 9c^2 \), is a difference of squares:
   \[
   (ab)^2 - (3c)^2
   \]
   This can be factored using the difference of squares formula:
   \[
   x^2 - y^2 = (x - y)(x + y)
   \]
   Applying this to the expression:
   \[
   a^2b^2 - 9c^2 = (ab - 3c)(ab + 3c)
   \]

3. **Final factored form:**
   Now, substitute this back into the original expression:
   \[
   d(a^2b^2 - 9c^2) = d(ab - 3c)(ab + 3c)
   \]

Thus, the fully factored form is:
\[
d(ab - 3c)(ab + 3c)
\]

Would you like further clarification on any step?

Here are 5 related questions you could explore:
1. How do you factor higher-degree polynomials?
2. What is the difference of cubes, and how is it factored?
3. Can any expression involving squares be factored using the difference of squares?
4. How do you factor trinomials of the form \( ax^2 + bx + c \)?
5. How do common factors help simplify expressions before factoring?

**Tip:** Always check for common factors before factoring any expression; it simplifies the problem significantly.

Learn how to factor the expression a²b²d - 9c²d by identifying the common factor and using the difference of squares formula. This step-by-step solution simplifies the process for high school students, covering key algebraic concepts such as factoring and the difference of squares theorem.

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