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The problem you're dealing with involves multiplying two binomials: \( (5y^2 - 6y)(7y^7 - y^3) \). The table given appears to be a setup for using the distributive property (FOIL or a grid method) to organize the multiplication.

Let's break down each part of the multiplication:

### Step 1: Distribute each term.
- First, distribute \( 5y^2 \) to both terms in \( (7y^7 - y^3) \):
  - \( 5y^2 \cdot 7y^7 = 35y^9 \)
  - \( 5y^2 \cdot (-y^3) = -5y^5 \)
  
- Then, distribute \( -6y \) to both terms in \( (7y^7 - y^3) \):
  - \( -6y \cdot 7y^7 = -42y^8 \)
  - \( -6y \cdot (-y^3) = 6y^4 \)

### Step 2: Fill in the table with the results.
Using the grid:
- The top left box will be \( 35y^9 \) (from \( 5y^2 \times 7y^7 \)).
- The top right box will be \( -5y^5 \) (from \( 5y^2 \times -y^3 \)).
- The bottom left box will be \( -42y^8 \) (from \( -6y \times 7y^7 \)).
- The bottom right box will be \( 6y^4 \) (from \( -6y \times -y^3 \)).

### Step 3: Combine the terms.
The final product is the sum of all the terms:
\[
35y^9 - 42y^8 - 5y^5 + 6y^4
\]
This is already in descending order of exponents.

Would you like further details on this or have any other questions?

Here are five related questions to expand on the concept:
1. How does the distributive property apply to multiplying polynomials in general?
2. What is the degree of the final polynomial result from this multiplication?
3. How do you simplify terms when distributing in polynomial multiplication?
4. Can you explain the significance of combining like terms in more complex expressions?
5. How would the method change if we were multiplying trinomials instead of binomials?

**Tip:** Always organize terms in descending order of exponents for clarity when working with polynomials.

Complete the table to find the product of (5y^2 - 6y)(7y^7 - y^3). Provide your answer in descending order of exponents.

Find the product of the binomials (5y^2 - 6y)(7y^7 - y^3) using the distributive property. The solution shows step-by-step multiplication and combining of terms in descending order of exponents. Suitable for students in Grades 8-10.

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