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To factor the polynomial \( 5x^4 + 20x^3 - 105x^2 \), let's follow these steps:

1. **Identify the Greatest Common Factor (GCF):**
   The GCF of \( 5x^4 \), \( 20x^3 \), and \( -105x^2 \) is \( 5x^2 \).

2. **Factor out the GCF:**
   \[
   5x^4 + 20x^3 - 105x^2 = 5x^2(x^2 + 4x - 21)
   \]

3. **Factor the quadratic expression \( x^2 + 4x - 21 \):**
   We need two numbers that multiply to \( -21 \) and add up to \( 4 \). These numbers are \( 7 \) and \( -3 \).

4. **Write the quadratic as a product of two binomials:**
   \[
   x^2 + 4x - 21 = (x + 7)(x - 3)
   \]

5. **Combine all factors:**
   \[
   5x^2(x + 7)(x - 3)
   \]

So, the completely factored form of the polynomial is \( 5x^2(x + 7)(x - 3) \).

**Correct answer:**
- **B.** \( 5x^2(x - 3)(x + 7) \)

Would you like further details or have any questions?

### Related Questions:
1. How do you find the greatest common factor (GCF) of a polynomial?
2. What methods can be used to factor a quadratic expression?
3. How can you check if your factorization is correct?
4. What is the difference between factoring completely and partially factoring a polynomial?
5. How do you factor polynomials with higher degrees (greater than 2)?

**Tip:** Always look for a greatest common factor (GCF) first when factoring polynomials. This simplifies the process and reduces the polynomial to a more manageable form.

What is the completely factored form of this polynomial? 5x^4 + 20x^3 - 105x^2

Learn how to factor the polynomial 5x^4 + 20x^3 - 105x^2. This step-by-step solution covers how to factor out the greatest common factor (GCF) and break down the quadratic expression, ideal for high school algebra students.

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