The expression you've provided is:

\[ 48x^4 + 8x^3 + 120x^2 + 20x \]

### Step 1: Factor out the Greatest Common Factor (GCF)

First, identify the GCF of all the terms. Here, the GCF is \( 4x \):

\[ 4x(12x^3 + 2x^2 + 30x + 5) \]

### Step 2: Check for Further Factorization

Now, we analyze the remaining polynomial \( 12x^3 + 2x^2 + 30x + 5 \). This polynomial doesn't factor further over the integers.

### Final Factored Form

The fully factored form of the given expression is:

\[ \boxed{4x(12x^3 + 2x^2 + 30x + 5)} \]

Would you like a detailed explanation of how to find the GCF or explore factorization methods for cubic polynomials? Here are some related questions:

1. How do you factor polynomials of higher degrees?
2. What is the importance of the greatest common factor (GCF) in factorization?
3. How do you factor cubic polynomials when they do not easily factor over integers?
4. What techniques can be used to simplify polynomials before factoring?
5. How do you determine if a polynomial is prime (i.e., cannot be factored further)?

**Tip:** Always start by factoring out the greatest common factor (GCF) before attempting other methods of factorization. This simplifies the problem and can make other factorization techniques easier to apply.

Learn how to factorize the polynomial 48x^4 + 8x^3 + 120x^2 + 20x step-by-step using the Greatest Common Factor (GCF) method. This problem involves polynomial factorization and understanding cubic polynomials. Suitable for students in Grades 10-12.

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