Sure, let's use the box method to distribute and simplify the polynomial [tex]$(4x - 6)(5x - 4)$[/tex].
First, we will create a table (box) to organize and compute the products of each term from the binomials. We will place the terms of the binomials along the top and left side of the box.
```
| 5x | -4
-----------------
4x | |
-----------------
-6 | |
```
Next, we fill in each cell of the box by multiplying the term at the top of the column with the term at the left of the row:
1. The cell in the top-left, where we multiply [tex]$4x$[/tex] and [tex]$5x$[/tex]:
[tex]\[ 4x \cdot 5x = 20x^2 \][/tex]
2. The cell in the top-right, where we multiply [tex]$4x$[/tex] and [tex]$-4$[/tex]:
[tex]\[ 4x \cdot (-4) = -16x \][/tex]
3. The cell in the bottom-left, where we multiply [tex]$-6$[/tex] and [tex]$5x$[/tex]:
[tex]\[ -6 \cdot 5x = -30x \][/tex]
4. The cell in the bottom-right, where we multiply [tex]$-6$[/tex] and [tex]$-4$[/tex]:
[tex]\[ -6 \cdot (-4) = 24 \][/tex]
Our table now looks like this:
```
| 5x | -4
---------------------
4x | 20x^2 | -16x
---------------------
-6 | -30x | 24
```
Now, sum up all the products inside the table:
1. [tex]$20x^2$[/tex] (from the top-left cell)
2. [tex]$-16x$[/tex] (from the top-right cell)
3. [tex]$-30x$[/tex] (from the bottom-left cell)
4. [tex]$24$[/tex] (from the bottom-right cell)
Combine the like terms (the terms involving [tex]$x$[/tex]):
[tex]\[ -16x + (-30x) = -46x \][/tex]
So, the simplified expression is:
[tex]\[ 20x^2 - 46x + 24 \][/tex]
Thus, the result of distributing and simplifying [tex]$(4x - 6)(5x - 4)$[/tex] using the box method is:
[tex]\[ 20x^2 - 46x + 24 \][/tex]
