https://file.aiquickdraw.com/mathbot/file/17260758771699vz5vcwh.jpg

The expression provided in the image is:

\[
-7x^2 + 4v - xv + 28x
\]

We are asked to factor this by grouping.

### Step-by-Step Solution:
1. **Group terms**: 
   \[
   (-7x^2 + 28x) + (- xv + 4v)
   \]

2. **Factor out the greatest common factor from each group**:
   - From \(-7x^2 + 28x\), factor out \(-7x\):
     \[
     -7x(x - 4)
     \]
   - From \(-xv + 4v\), factor out \(v\):
     \[
     v(-x + 4)
     \]

3. **Rewrite the expression**:
   \[
   -7x(x - 4) + v(-x + 4)
   \]

4. **Notice that \((x - 4)\) and \((-x + 4)\) are the same up to a sign. To factor completely, factor out \(-(x - 4)\)**:
   \[
   (-7x + v)(-x + 4)
   \]

This matches the factored form provided in the box of the image, which is \((7x + v)(-x + 4)\), except with the signs accounted for in the process.

Let me know if you want further clarification on this or have any other questions!

### 5 Related Questions:
1. How do you determine the greatest common factor when factoring by grouping?
2. Can factoring by grouping be used for any polynomial, or are there specific types of polynomials where it's more useful?
3. What other methods can be used to factor quadratics besides grouping?
4. How do you handle negative signs in factoring by grouping?
5. Why is it important to check for common factors before grouping?

### Tip:
Always double-check your factored terms by expanding to ensure you get back the original expression!

Factor by grouping: -7x^2 + 4v - xv + 28x

Learn how to factor the polynomial -7x^2 + 4v - xv + 28x by grouping. This step-by-step solution involves grouping terms, factoring out common factors, and applying the distributive property to get the final answer. Suitable for high school students in Grades 9-12.

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