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Part b of question 3 asks you to determine the length and width of a rectangular sign by using the value \( x = 5 \) meters in the binomials obtained from the factorization of the trinomial \( x^2 - 6x - 7 \) from part a.

Let's break this down step by step:

### Part a Recap:
- The trinomial provided is \( x^2 - 6x - 7 \).
- The goal was to factor this trinomial. The factorized form is \( (x - 7)(x + 1) \).
  - Here, \( x - 7 \) and \( x + 1 \) represent the possible length and width of the sign, depending on which you assign to each.

### Part b Explanation:
1. **Substitute \( x = 5 \) into the binomials**:
   - Length: \( x - 7 = 5 - 7 = -2 \) meters
   - Width: \( x + 1 = 5 + 1 = 6 \) meters

2. **Interpretation**:
   - In reality, a physical dimension cannot be negative. Therefore, it seems there's a logical issue with the factor assignment.
   - If we swap the assignment:
     - Length: \( x + 1 = 6 \) meters
     - Width: \( x - 7 = -2 \) meters

However, as we cannot have negative physical dimensions, this might indicate that the factorization step requires revisiting. For now, the question might be implying that the width is 6 meters and possibly there's an error in the problem setup.

### Conclusion:
- **Length**: 6 meters (from \( x + 1 \))
- **Width**: Not directly usable since \( x - 7 \) gives a negative value. 

### Do you want to revisit part a's factorization? Or perhaps review the overall problem setup?

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Would you like more details on any specific part or have any further questions? Here are some related questions:

1. How do we ensure factorization is correct for a given quadratic trinomial?
2. Why can’t dimensions of a physical object be negative?
3. Can the quadratic equation \( x^2 - 6x - 7 \) represent something other than a rectangle with the given context?
4. What are the alternative methods to factorize quadratic expressions?
5. How does the quadratic formula relate to factorization in this context?

**Tip:** When dealing with physical quantities, always ensure that the results you calculate make sense within the real-world context of the problem.

In this problem, we factorize the trinomial x^2 - 6x - 7 to represent the dimensions of a rectangular sign. The solution involves using the value x = 5 meters to determine the length and width, with a focus on interpreting physical constraints. Suitable for Grades 8-10.

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