To multiply the two binomials \((6v - 4w + 7)\) and \((2v - 5w - 1)\), we will use the distributive property (often referred to as the FOIL method when dealing with binomials), but since these are trinomials, we’ll distribute each term from the first trinomial to each term in the second trinomial.

\[
(6v - 4w + 7)(2v - 5w - 1)
\]

Step 1: Multiply each term in the first set of parentheses by each term in the second set of parentheses.

1. \(6v \cdot 2v = 12v^2\)
2. \(6v \cdot (-5w) = -30vw\)
3. \(6v \cdot (-1) = -6v\)
4. \(-4w \cdot 2v = -8vw\)
5. \(-4w \cdot (-5w) = 20w^2\)
6. \(-4w \cdot (-1) = 4w\)
7. \(7 \cdot 2v = 14v\)
8. \(7 \cdot (-5w) = -35w\)
9. \(7 \cdot (-1) = -7\)

Step 2: Combine like terms.

- For \(v^2\): \(12v^2\)
- For \(vw\): \(-30vw - 8vw = -38vw\)
- For \(v\): \(-6v + 14v = 8v\)
- For \(w^2\): \(20w^2\)
- For \(w\): \(4w - 35w = -31w\)
- Constant: \(-7\)

Final result:

\[
12v^2 - 38vw + 8v + 20w^2 - 31w - 7
\]

Would you like more details on any of these steps?

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**Relative questions:**

1. How do you apply the distributive property to multiply polynomials?
2. What is the process for simplifying like terms in polynomial expressions?
3. Can you explain how to identify like terms when dealing with multiple variables?
4. What are common mistakes to avoid when multiplying polynomials?
5. How do you factor polynomials after multiplying them?

**Tip:** When combining like terms, be careful to match both the variable and the exponent to avoid errors.

Learn how to multiply the polynomials (6v - 4w + 7) and (2v - 5w - 1) step-by-step. This problem uses the distributive property and shows how to combine like terms. Ideal for students in Grades 9-12.

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