Question

Use the area method to fill in the boxes and provide the product of the binomials: a) (4𝑥 + 3)(2𝑥 − 5) i) Example: (1) (2) (3) ii) Use the caret sign (^) to denote an exponent: x² = x^2 (1) The caret sign on your keyboard is to hold SHIFT + 6 Fill in the green portions of the box below and provide the simplified product. (4𝑥 + 3)(2𝑥 − 5) 4x 3 2x -5 Simplified solution:

Answer 1

To use the area method to find the product of the binomials \((4x + 3)(2x - 5)\), we can set up a box diagram that represents the components of each binomial.

We will fill in the box as follows:

- The top side of the box will be split into parts for each term in the first binomial \(4x + 3\).
- The left side of the box will be split into parts for each term in the second binomial \(2x - 5\).

Here's how it looks:

```
| 2x | -5 |
--------------------------------
4x | 8x² | -20x |
--------------------------------
3 | 6x | -15 |
```

Now let's fill in each box:

1. **Top Left Box**: Multiply \(4x\) by \(2x\):
\[
4x \cdot 2x = 8x^2
\]

2. **Top Right Box**: Multiply \(4x\) by \(-5\):
\[
4x \cdot (-5) = -20x
\]

3. **Bottom Left Box**: Multiply \(3\) by \(2x\):
\[
3 \cdot 2x = 6x
\]

4. **Bottom Right Box**: Multiply \(3\) by \(-5\):
\[
3 \cdot (-5) = -15
\]

### Now we can combine all the parts:

The product of the two binomials is obtained by adding all the values from the boxes together:

\[
8x^2 - 20x + 6x - 15
\]

Now, combine like terms (\(-20x + 6x\)):

\[
8x^2 - 14x - 15
\]

### Simplified Solution

The simplified product of \((4x + 3)(2x - 5)\) is:
\[
8x^2 - 14x - 15
\]