To multiply the expressions [tex]\( (6x - y)(2x - y + 2) \)[/tex], let's use the distributive property and multiply each term in the first expression by each term in the second expression. Here's a detailed, step-by-step solution:
1. Start distributing [tex]\( 6x \)[/tex] over each term in the expression [tex]\( 2x - y + 2 \)[/tex]:
[tex]\[
6x * 2x = 12x^2
\][/tex]
[tex]\[
6x * (-y) = -6xy
\][/tex]
[tex]\[
6x * 2 = 12x
\][/tex]
2. Next distribute [tex]\( -y \)[/tex] over each term in the expression [tex]\( 2x - y + 2 \)[/tex]:
[tex]\[
-y * 2x = -2xy
\][/tex]
[tex]\[
-y * (-y) = y^2
\][/tex]
[tex]\[
-y * 2 = -2y
\][/tex]
3. Combine all products to get the expression before combining like terms:
[tex]\[
12x^2 - 6xy + 12x - 2xy + y^2 - 2y
\][/tex]
4. Combine like terms:
[tex]\[
12x^2 - 6xy - 2xy + 12x + y^2 - 2y
\][/tex]
[tex]\[
12x^2 - 8xy + 12x + y^2 - 2y
\][/tex]
Therefore, the final answer is:
[tex]\[
12x^2 - 8xy + 12x + y^2 - 2y
\][/tex]
Now, let's place each term in its correct spot in the table:
\begin{tabular}{|c|c|c|c|}
\hline
[tex]$y^2$[/tex] & [tex]$-6xy$[/tex] & [tex]$-2xy$[/tex] & [tex]$12x^2$[/tex] \\
\hline
[tex]$-2y$[/tex] & [tex]$12x$[/tex] & & \\
\hline
\end{tabular}
\begin{tabular}{|c|c|c|c|}
\hline
& [tex]$2x$[/tex] & [tex]$-y$[/tex] & 2 \\
\hline
[tex]$6x$[/tex] & [tex]$12x^2$[/tex] & [tex]$-6xy$[/tex] & [tex]$12x$[/tex] \\
\hline
[tex]$-y$[/tex] & [tex]$-2xy$[/tex] & [tex]$y^2$[/tex] & [tex]$-2y$[/tex] \\
\hline
\end{tabular}
