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Let's factor the given expression completely.

The expression is:

\[
12x^5 - 27x^3
\]

### Step-by-step solution:

1. **Find the greatest common factor (GCF):**  
   The GCF of \(12x^5\) and \(27x^3\) involves the numerical coefficients and the powers of \(x\).
   - The GCF of 12 and 27 is 3.
   - The GCF of \(x^5\) and \(x^3\) is \(x^3\).

   Thus, the GCF of the entire expression is \(3x^3\).

2. **Factor out the GCF:**  
   We can factor out \(3x^3\) from both terms.

   \[
   12x^5 - 27x^3 = 3x^3(4x^2 - 9)
   \]

3. **Factor the remaining quadratic expression:**  
   Now, we need to check if \(4x^2 - 9\) can be factored further. Notice that this is a **difference of squares**:

   \[
   4x^2 - 9 = (2x - 3)(2x + 3)
   \]

4. **Final factored form:**  
   Substituting back, the fully factored expression is:

   \[
   12x^5 - 27x^3 = 3x^3(2x - 3)(2x + 3)
   \]

This is the completely factored form of the expression.

---

Would you like further details on any step or have additional questions?

Here are 5 related questions for you to explore:

1. How do you factor the difference of squares in general?
2. What is the process to find the greatest common factor (GCF) of two terms?
3. How do you check if a polynomial can be factored further?
4. Can a sum of squares be factored over the real numbers?
5. What are some other types of special polynomial factorizations?

**Tip:** Always check for common factors before attempting more complex factoring methods.

This problem involves factoring the expression 12x^5 - 27x^3 completely. The solution utilizes the greatest common factor (GCF) method and the difference of squares formula, making it suitable for students in Grades 9-11.

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