Sure, let's use the box method (also known as the area model) to distribute and simplify [tex]\((-6x + 5)(-6x - 6)\)[/tex].
First, set up the box by writing the two binomial expressions along the top and side:
[tex]\[
\begin{array}{c|c|c|}
& -6x & -6 \\
\hline
-6x & & \\
\hline
5 & & \\
\hline
\end{array}
\][/tex]
Now, fill in each cell by multiplying the corresponding terms from the top row and the left column.
### Step-by-Step Calculations:
1. Calculate the product of [tex]\((-6x) \cdot (-6x)\)[/tex]:
[tex]\[ (-6x) \cdot (-6x) = 36x^2 \][/tex]
2. Calculate the product of [tex]\((-6x) \cdot (-6)\)[/tex]:
[tex]\[ (-6x) \cdot (-6) = 36x \][/tex]
3. Calculate the product of [tex]\(5 \cdot (-6x)\)[/tex]:
[tex]\[ 5 \cdot (-6x) = -30x \][/tex]
4. Calculate the product of [tex]\(5 \cdot (-6)\)[/tex]:
[tex]\[ 5 \cdot (-6) = -30 \][/tex]
Now, place these products into the appropriate cells of the box:
[tex]\[
\begin{array}{c|c|c|}
& -6x & -6 \\
\hline
-6x & 36x^2 & 36x \\
\hline
5 & -30x & -30 \\
\hline
\end{array}
\][/tex]
### Summarize:
Finally, add all the terms together:
[tex]\[
36x^2 + 36x - 30x - 30
\][/tex]
Combine like terms:
[tex]\[
36x^2 + (36x - 30x) - 30 = 36x^2 + 6x - 30
\][/tex]
So, the simplified form of the expression [tex]\((-6x + 5)(-6x - 6)\)[/tex] is:
[tex]\[
36x^2 + 6x - 30
\][/tex]
