Use the box method to distribute and simplify left parenthesis, minus, 6, x, plus, 1, right parenthesis, left parenthesis, 3, x, plus, 5, right parenthesis, .(−6*x*+1)(3*x*+5). Drag and drop the terms to the correct locations of the table.

To distribute and simplify the expression \((-6x + 1)(3x + 5)\) using the box method, we follow these steps:

1. **Set up the box**: Create a 2x2 grid (since each binomial has two terms).
2. **Label the rows and columns**: Label the rows with the terms from \(-6x + 1\) and the columns with the terms from \(3x + 5\).

Here's how the grid looks:

|             |  \(3x\) | \(5\)   |
|-------------|---------|---------|
| \(-6x\)     |         |         |
| \(1\)       |         |         |

3. **Multiply the terms**: Fill in each cell with the product of the corresponding row and column terms.

- Top left cell: \((-6x) \times (3x)\)
- Top right cell: \((-6x) \times (5)\)
- Bottom left cell: \(1 \times (3x)\)
- Bottom right cell: \(1 \times (5)\)

Now, fill in the table:

|             |  \(3x\)      | \(5\)    |
|-------------|--------------|----------|
| \(-6x\)     | \(-18x^2\)   | \(-30x\) |
| \(1\)       | \(3x\)       | \(5\)    |

4. **Combine like terms**: The expression from the box is:
   \[
   -18x^2 - 30x + 3x + 5
   \]

Combine the like terms:

   \[
   -18x^2 - 27x + 5
   \]

So, the simplified expression is \(-18x^2 - 27x + 5\).

Would you like more details or have any questions?

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**Related Questions:**
1. What is the box method, and how does it differ from other distribution methods?
2. Can you explain how to set up a box for a trinomial times binomial multiplication?
3. How does the box method help in understanding the distributive property?
4. Can the box method be used for polynomial division? If so, how?
5. What are some common mistakes to avoid when using the box method?
6. How does the box method handle negative signs and coefficients?
7. Can the box method be applied to expressions with more than two terms in each factor?
8. How do the results from the box method compare to those obtained using the FOIL method?

**Tip:** When using the box method, always double-check the signs of the terms to ensure correct distribution.

Learn how to distribute and simplify the expression (-6x + 1)(3x + 5) using the box method with step-by-step instructions. Understand polynomial multiplication and the distributive property in algebra. Suitable for Grades 7-9 students.

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