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The given polynomial is \( uy + 7u - 5y - 35 \). To factor this expression by grouping, we follow these steps:

### Step-by-step process:

1. **Group terms:**
   \[
   (uy + 7u) - (5y + 35)
   \]
   
2. **Factor out common factors from each group:**
   - From the first group \( uy + 7u \), factor out \( u \):
     \[
     u(y + 7)
     \]
   - From the second group \( -5y - 35 \), factor out \( -5 \):
     \[
     -5(y + 7)
     \]
   
3. **Factor out the common binomial factor \( (y + 7) \):**
   \[
   (y + 7)(u - 5)
   \]

### Final Factored Form:
\[
(y + 7)(u - 5)
\]

This is the fully factored form of the given polynomial.

Would you like further explanation or have any questions?

Here are some related questions for further practice:
1. How would you apply grouping for a four-term polynomial?
2. Can every four-term polynomial be factored by grouping?
3. What would happen if there were no common factors between groups?
4. How do you check if the factored form is correct?
5. What are some common factoring techniques for multivariable polynomials?

**Tip:** Always check if the terms in a polynomial can be rearranged to form groups that can be factored easily.

Factor the polynomial uy + 7u - 5y - 35 by grouping.

Learn how to factor the polynomial uy + 7u - 5y - 35 by grouping. This problem involves recognizing common factors and applying the distributive property. Suitable for students in Grades 8-10, covering key algebraic concepts like factoring polynomials.

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