use the area method to fill in the boses and provide the product of binomials (4x+3)(2x-5)

Answer 1

To use the area method to multiply the binomials \((4x + 3)(2x - 5)\), we can set up a box (or area) diagram. Here’s how you can visualize it:

Step 1: Create a 2x2 Box

We will create a box divided into 4 smaller boxes since we have two terms in the first binomial and two terms in the second binomial.

Step 2: Label the Rows and Columns

• Row 1: Label the first box with \(4x\) and the second box with \(3\).
• Column 1: Label the first box with \(2x\) and the second box with \(-5\).

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

Step 3: Calculate Each Box’s Area

Now we calculate the area of each smaller box:

1. Top left box: \(4x \times 2x = 8x^2\)
2. Top right box: \(4x \times -5 = -20x\)
3. Bottom left box: \(3 \times 2x = 6x\)
4. Bottom right box: \(3 \times -5 = -15\)

Step 4: Write Down the Products

Now, we write down each area:

• \(8x^2\)
• \(-20x\)
• \(6x\)
• \(-15\)

Step 5: Combine Like Terms

Now, sum all the areas:

\[ 8x^2 + (-20x) + 6x + (-15) = 8x^2 - 14x - 15 \]

Final Result

Therefore, the product of the binomials \((4x + 3)(2x - 5)\) is:

\[ \boxed{8x^2 - 14x - 15} \]