To use the box method (also known as the area method) to distribute and simplify [tex]\((4x - 6)(5x - 4)\)[/tex], follow these steps:
### Step 1: Set up a 2x2 grid.
We will make a table. Label the columns with [tex]\(4x\)[/tex] and [tex]\(-6\)[/tex], and label the rows with [tex]\(5x\)[/tex] and [tex]\(-4\)[/tex].
### Step 2: Distribute each term to the appropriate box.
We will multiply each term in the first binomial by each term in the second binomial and place the result in the corresponding cell of our grid.
### Grid and Multiplications
1. [tex]\( (4x) \times (5x) \)[/tex]: Place [tex]\(20x^2\)[/tex] in the top-left cell.
2. [tex]\( (4x) \times (-4) \)[/tex]: Place [tex]\(-16x\)[/tex] in the top-right cell.
3. [tex]\( (-6) \times (5x) \)[/tex]: Place [tex]\(-30x\)[/tex] in the bottom-left cell.
4. [tex]\( (-6) \times (-4) \)[/tex]: Place [tex]\(24\)[/tex] in the bottom-right cell.
Here's how the table looks:
[tex]\[
\begin{array}{c|c|c}
& 4x & -6 \\
\hline
5x & 20x^2 & -16x \\
\hline
-4 & -30x & 24 \\
\end{array}
\][/tex]
### Step 3: Combine like terms.
Now, add up all the terms from the boxes:
- [tex]\(20x^2\)[/tex] (from the top-left box)
- [tex]\(-16x\)[/tex] (from the top-right box)
- [tex]\(-30x\)[/tex] (from the bottom-left box)
- [tex]\(24\)[/tex] (from the bottom-right box)
Combine the terms:
[tex]\[
20x^2 + (-16x) + (-30x) + 24 = 20x^2 - 46x + 24
\][/tex]
### Final Answer
[tex]\((4x - 6)(5x - 4) = 20x^2 - 46x + 24\)[/tex]
So, the expanded and simplified form of [tex]\((4x - 6)(5x - 4)\)[/tex] is [tex]\(20x^2 - 46x + 24\)[/tex].
