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Let's solve the problem step by step:

### Problem Breakdown

The task requires finding the greatest common factor (GCF) of pairs of expressions and then identifying the corresponding numbers on a grid.

1. **Find the greatest common factor (GCF) for each pair of expressions.**
2. **Circle the answer on the grid.**
3. **Combine the answers to form the largest 3-digit number possible, which will be the lock code.**

### Solving the Pairs

#### 1) \( 14x^2 \) and \( 28x \)
- **Factor each term:**
  - \( 14x^2 = 2 \times 7 \times x \times x \)
  - \( 28x = 2 \times 14 \times x = 2 \times 7 \times 2 \times x \)

- **Common factors:**
  - The common factors are \( 2, 7, \) and \( x \).

- **GCF:**
  - \( 2 \times 7 \times x = 14x \)

**Answer: \( 14x \)**

#### 2) \( 15xy^2 \) and \( 25x^2y^2 \)
- **Factor each term:**
  - \( 15xy^2 = 3 \times 5 \times x \times y \times y \)
  - \( 25x^2y^2 = 5 \times 5 \times x \times x \times y \times y \)

- **Common factors:**
  - The common factors are \( 5, x, y^2 \).

- **GCF:**
  - \( 5 \times x \times y^2 = 5xy^2 \)

**Answer: \( 5xy^2 \)**

#### 3) \( 14x^2 \) and \( 21x \)
- **Factor each term:**
  - \( 14x^2 = 2 \times 7 \times x \times x \)
  - \( 21x = 3 \times 7 \times x \)

- **Common factors:**
  - The common factors are \( 7 \) and \( x \).

- **GCF:**
  - \( 7 \times x = 7x \)

**Answer: \( 7x \)**

### Corresponding Numbers on the Grid

- **14x corresponds to 4.**
- **5xy^2 corresponds to 6.**
- **7x corresponds to 1.**

### Form the Largest 3-Digit Number

- The largest number that can be formed with the digits 4, 6, and 1 is \( 641 \).

### Lock Code

**The lock code is \( 641 \).**

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Do you want more details or have any questions? Here are some related questions to further explore the topic:

1. What is the process for finding the GCF of polynomial expressions?
2. Can the GCF of a pair of monomials ever be greater than the smaller of the two terms?
3. How do you factor a polynomial to find its greatest common factor?
4. What are the common mistakes to avoid when finding the GCF of polynomials?
5. How can you verify that you have found the correct GCF?
6. What is the significance of the GCF in simplifying rational expressions?
7. How does finding the GCF help in solving polynomial equations?
8. Can the method of finding the GCF be applied to multi-variable polynomials?

**Tip:** When finding the greatest common factor, always break down each term into its prime factors and variables, then identify the lowest power of each common factor.

Learn how to find the Greatest Common Factor (GCF) of polynomial expressions step-by-step. Solve problems involving polynomial GCFs and understand how to apply prime factorization to identify common factors. Suitable for high school students studying algebra and polynomial equations.

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