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To solve this problem, let's analyze each option to see if it correctly represents the factorization of the given expression.

### Step-by-Step Analysis:

1. **Option A**: \( x^2 - x + 49 = (x - 7)^2 \)
   - Expanding \( (x - 7)^2 \) gives:
     \[
     (x - 7)^2 = x^2 - 14x + 49
     \]
   - This does **not** match \( x^2 - x + 49 \), so **Option A is incorrect**.

2. **Option B**: \( x^2 - 3x - 28 = (x - 7)(x + 4) \)
   - Expanding \( (x - 7)(x + 4) \):
     \[
     (x - 7)(x + 4) = x^2 + 4x - 7x - 28 = x^2 - 3x - 28
     \]
   - This matches the given expression \( x^2 - 3x - 28 \), so **Option B is correct**.

3. **Option C**: \( x^2 + 25 = (x + 5)^2 \)
   - Expanding \( (x + 5)^2 \):
     \[
     (x + 5)^2 = x^2 + 10x + 25
     \]
   - This does **not** match \( x^2 + 25 \), so **Option C is incorrect**.

4. **Option D**: \( x^2 - 3x - 4 = (x + 4)(x + 1) \)
   - Expanding \( (x + 4)(x + 1) \):
     \[
     (x + 4)(x + 1) = x^2 + x + 4x + 4 = x^2 + 5x + 4
     \]
   - This does **not** match \( x^2 - 3x - 4 \), so **Option D is incorrect**.

### Conclusion:
The correct answer is **Option B**: \( x^2 - 3x - 28 = (x - 7)(x + 4) \).

Would you like further explanation or details on this solution?

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Here are some related questions to deepen understanding:

1. What is the general approach to factor quadratic expressions?
2. How can we confirm that a factorization is correct?
3. Why does \( x^2 + 25 \) not factor as \( (x + 5)^2 \)?
4. What are some techniques for factoring expressions like \( x^2 - 3x - 4 \)?
5. How do the signs in a quadratic expression affect its factorization?

**Tip:** When factoring, always expand back to check if you arrive at the original expression. This helps confirm accuracy.

Select the true factorization for the expression given in the options.

Learn how to correctly factorize the quadratic expression x^2 - 3x - 28. This detailed solution covers key algebraic concepts and verifies factorization through expansion. Ideal for students in Grades 8-10.

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