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