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The expression to factor completely is:

\[
4x^4 - 16x^3 - 180x^2
\]

### Step-by-step solution:

1. **Factor out the greatest common factor (GCF):**
   The GCF of \(4x^4\), \(-16x^3\), and \(-180x^2\) is \(4x^2\).

   Factor out \(4x^2\):

   \[
   4x^2(x^2 - 4x - 45)
   \]

2. **Factor the quadratic inside the parentheses:**
   We need to factor \(x^2 - 4x - 45\). 

   To factor this, find two numbers that multiply to \(-45\) and add up to \(-4\). These numbers are \(-9\) and \(5\), since \((-9)(5) = -45\) and \((-9) + 5 = -4\).

   Therefore, the quadratic can be factored as:

   \[
   (x - 9)(x + 5)
   \]

3. **Write the complete factorization:**

   \[
   4x^2(x - 9)(x + 5)
   \]

Thus, the fully factored form of \(4x^4 - 16x^3 - 180x^2\) is:

\[
4x^2(x - 9)(x + 5)
\]

Would you like more details or have any questions?

### Related Questions:
1. How do you find the greatest common factor (GCF) of multiple terms?
2. What is the method for factoring quadratic expressions?
3. How do you check if a factorization is correct?
4. What role does factoring play in solving polynomial equations?
5. How can factoring help simplify polynomial division?

### Tip:
Always start by factoring out the greatest common factor (GCF) first before attempting more complex factoring methods.

Learn how to completely factor the polynomial expression 4x^4 - 16x^3 - 180x^2. This step-by-step guide covers algebraic factoring, factoring out the greatest common factor (GCF), and quadratic factorization. Suitable for high school algebra students.

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