Certainly! Let's use the box method to distribute and simplify [tex]\((5x + 1)(-2x - 5)\)[/tex].
1. First, let's set up the box:
[tex]\[
\begin{array}{|c|c|c|}
\hline
& -2x & -5 \\
\hline
5x & & \\
\hline
1 & & \\
\hline
\end{array}
\][/tex]
2. Now, let's fill in each cell of the box by multiplying the terms corresponding to the row and column.
- For the cell at the intersection of [tex]\(5x\)[/tex] (the first row) and [tex]\(-2x\)[/tex] (the first column):
[tex]\[
5x \times (-2x) = -10x^2
\][/tex]
- For the cell at the intersection of [tex]\(5x\)[/tex] (the first row) and [tex]\(-5\)[/tex] (the second column):
[tex]\[
5x \times (-5) = -25x
\][/tex]
- For the cell at the intersection of [tex]\(1\)[/tex] (the second row) and [tex]\(-2x\)[/tex] (the first column):
[tex]\[
1 \times (-2x) = -2x
\][/tex]
- For the cell at the intersection of [tex]\(1\)[/tex] (the second row) and [tex]\(-5\)[/tex] (the second column):
[tex]\[
1 \times (-5) = -5
\][/tex]
3. Now the box looks like this:
[tex]\[
\begin{array}{|c|c|c|}
\hline
& -2x & -5 \\
\hline
5x & -10x^2 & -25x \\
\hline
1 & -2x & -5 \\
\hline
\end{array}
\][/tex]
4. Next, combine like terms from the results obtained in the box. Here, we have:
- The coefficient of [tex]\(x^2\)[/tex] is [tex]\(-10\)[/tex].
- The coefficients of [tex]\(x\)[/tex] terms are [tex]\(-25x\)[/tex] and [tex]\(-2x\)[/tex], which combine to [tex]\(-27x\)[/tex].
- The constant term is [tex]\(-5\)[/tex].
5. Therefore, combining everything together:
[tex]\((5x + 1)(-2x - 5) = -10x^2 - 27x - 5\)[/tex]
To summarize:
[tex]\[
(5x + 1)(-2x - 5) = -10x^2 - 27x - 5
\][/tex]
