Let's factor the expression \( 48x^4 + 8x^3 + 120x^2 + 20x \).

### Step 1: Identify the Greatest Common Factor (GCF)
First, let's find the GCF of all the terms in the expression.

- The coefficients are \( 48, 8, 120, \) and \( 20 \).
- The GCF of these coefficients is \( 4 \).
- Each term also has a factor of \( x \).

So, the GCF of the entire expression is \( 4x \).

### Step 2: Factor out the GCF
Now, factor \( 4x \) out of each term:

\[
48x^4 + 8x^3 + 120x^2 + 20x = 4x(12x^3 + 2x^2 + 30x + 5)
\]

### Step 3: Check for Further Factorization
Next, we'll check if the polynomial inside the parentheses \( 12x^3 + 2x^2 + 30x + 5 \) can be factored further.

#### Grouping Method:
Let's attempt to factor by grouping:

1. Group the terms: \((12x^3 + 2x^2) + (30x + 5)\).
2. Factor out the common factors from each group:
   \[
   2x^2(6x + 1) + 5(6x + 1)
   \]
3. Notice that \( (6x + 1) \) is a common factor:
   \[
   (2x^2 + 5)(6x + 1)
   \]

### Final Factored Form
So, the fully factored form of the original expression is:

\[
\boxed{4x(2x^2 + 5)(6x + 1)}
\]

Would you like to explore the factorization process further, or do you have any other questions? Here are some related questions you might find interesting:

1. How does the grouping method work for factoring polynomials?
2. What are other methods for factoring higher-degree polynomials?
3. How can you verify that a factorization is correct?
4. When should you check for complex roots during factorization?
5. How do you factor expressions that don’t have an easily identifiable GCF?

**Tip:** After factoring, always check your work by expanding the factors back out to ensure they multiply to the original expression.

Learn how to factor the polynomial 48x^4 + 8x^3 + 120x^2 + 20x with a step-by-step guide. This problem covers polynomial factorization, identifying the Greatest Common Factor (GCF), and factoring by grouping. Suitable for high school students studying algebra and polynomial mathematics.

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