To identify the incorrect product in the table and find the correct value, we need to carefully multiply each term in one polynomial by each term in the other polynomial. Let's go through this step-by-step.
Shana is multiplying [tex]\((2x + y)\)[/tex] with [tex]\((5x - y + 3)\)[/tex].
Here is the table provided by Shana:
[tex]\[
\begin{array}{|c|c|c|c|}
\hline
& 5x & -y & 3 \\
\hline
2x & 10x^2 & -2y & 6x \\
\hline
y & 5xy & -y^2 & 3y \\
\hline
\end{array}
\][/tex]
### Step-by-Step Multiplication:
1. Multiply [tex]\(2x\)[/tex] by each term in the second polynomial:
- [tex]\(2x \cdot 5x = 10x^2\)[/tex]
- [tex]\(2x \cdot (-y) = -2xy\)[/tex]
- [tex]\(2x \cdot 3 = 6x\)[/tex]
2. Multiply [tex]\(y\)[/tex] by each term in the second polynomial:
- [tex]\(y \cdot 5x = 5xy\)[/tex]
- [tex]\(y \cdot (-y) = -y^2\)[/tex]
- [tex]\(y \cdot 3 = 3y\)[/tex]
Let's put the results in the table:
[tex]\[
\begin{array}{|c|c|c|c|}
\hline
& 5x & -y & 3 \\
\hline
2x & 10x^2 & -2xy & 6x \\
\hline
y & 5xy & -y^2 & 3y \\
\hline
\end{array}
\][/tex]
### Compare Shana's Work with the Correct Multiplication:
- [tex]\(10x^2\)[/tex] matches with Shana's table.
- [tex]\(-2xy\)[/tex] does not match the [tex]\(-2y\)[/tex] in Shana's table.
- [tex]\(6x\)[/tex] matches with Shana's table.
- [tex]\(5xy\)[/tex] matches with Shana's table.
- [tex]\(-y^2\)[/tex] matches with Shana's table.
- [tex]\(3y\)[/tex] matches with Shana's table.
The incorrect product in Shana's table is [tex]\(-2y\)[/tex]. The correct product should be [tex]\(-2xy\)[/tex].
### Conclusion:
The value Shana should have written is [tex]\(-2xy\)[/tex].
