Let's simplify the expression \(24p^4 + 24p^2q - 27q^2\).

### Step 1: Factor out the Greatest Common Factor (GCF)
First, identify the GCF of the terms: \(24p^4\), \(24p^2q\), and \(-27q^2\). The GCF is 3.

Factoring out the GCF:
\[
3(8p^4 + 8p^2q - 9q^2)
\]

### Step 2: Factor the Quadratic Expression
Now, let's factor the quadratic expression \(8p^4 + 8p^2q - 9q^2\). To factor this, we can treat \(p^2\) as a single variable.

Rewrite \(8p^4 + 8p^2q - 9q^2\) in terms of \(p^2\):
\[
8(p^2)^2 + 8p^2q - 9q^2
\]

Let's try factoring by grouping or using the quadratic formula approach. 

This expression can be factored into:
\[
8(p^2 + q)(p^2 - \frac{9q}{8})
\]

### Final Factored Form
Putting it all together:
\[
3 \times (8(p^2 + q)(p^2 - \frac{9q}{8}))
\]

However, due to the fractional factor, the form can also be rewritten or left as:
\[
3(8p^4 + 8p^2q - 9q^2)
\]

Would you like more details or have any questions? Here are some related topics:

1. How to factor quadratic expressions with variable exponents.
2. Understanding factoring by grouping.
3. Exploring the quadratic formula for complex expressions.
4. Simplifying algebraic expressions with multiple variables.
5. Tips for identifying common factors in algebraic expressions.

**Tip:** Always start factoring by looking for the Greatest Common Factor (GCF) to simplify the expression before diving into more complex factoring methods!

Learn how to factor the polynomial expression 24p^4 + 24p^2q - 27q^2. This problem covers key algebraic concepts such as finding the Greatest Common Factor (GCF), factoring quadratics, and solving equations with variable exponents. Suitable for students in Grades 9-12.

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