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The image contains a question asking for the **GCF (Greatest Common Factor)** of the following expression:

\[
4x^2(x - 2)^3(x - 1) - 12x^2(x - 2)^2(x - 1)^2
\]

### Step-by-Step Solution:

#### 1. Factor out common terms:

The first term is:  
\[
4x^2(x - 2)^3(x - 1)
\]
The second term is:  
\[
12x^2(x - 2)^2(x - 1)^2
\]

Notice that both terms have the following common factors:
- \( x^2 \)
- \( (x - 2)^2 \)
- \( (x - 1) \)

#### 2. Identify the smallest powers of each common factor:
- The lowest power of \( x^2 \) is \( x^2 \) itself.
- The lowest power of \( (x - 2)^3 \) and \( (x - 2)^2 \) is \( (x - 2)^2 \).
- The lowest power of \( (x - 1)^1 \) and \( (x - 1)^2 \) is \( (x - 1) \).

#### 3. The GCF:
Thus, the GCF of the given expression is:
\[
x^2 (x - 2)^2 (x - 1)
\]

Would you like more details or have any further questions?

Here are 5 related questions to expand on this topic:
1. How would you simplify the expression after factoring out the GCF?
2. What is the process for finding the least common multiple (LCM) of two polynomials?
3. How does factoring polynomials help in solving equations?
4. Can GCF be applied to rational functions, and if so, how?
5. What role do exponents play in determining the GCF of polynomial expressions?

**Tip:** When factoring polynomials, always start by identifying common terms or factors, including constants and variable powers. This simplifies the process of finding the GCF.

What is the GCF (Greatest Common Factor) of the expression 4x^2(x - 2)^3(x - 1) - 12x^2(x - 2)^2(x - 1)^2?

Learn how to find the greatest common factor (GCF) of the polynomial expression 4x^2(x - 2)^3(x - 1) - 12x^2(x - 2)^2(x - 1)^2. This solution breaks down the steps, highlighting key algebraic concepts such as factoring and the properties of exponents. Ideal for students in Grades 9-11.

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