To determine the value of [tex]\( A \)[/tex] in the given table, we need to understand what [tex]\( A \)[/tex] represents. The table can be interpreted as a multiplication table where each cell is the product of the binomials from the headers of the respective rows and columns.
The headers for the first row and first column are:
- The values for the first row headers are [tex]\(3x\)[/tex] and [tex]\(5\)[/tex].
- The values for the first column headers are [tex]\(-x\)[/tex] and [tex]\(2\)[/tex].
To find [tex]\( A \)[/tex], we need to multiply the elements in the header row ([tex]\(3x\)[/tex]) and the header column ([tex]\(-x\)[/tex]) that correspond to the position of [tex]\( A \)[/tex]:
The element at this position had headers [tex]\( 3x \)[/tex] horizontally and [tex]\(-x\)[/tex] vertically.
So,
[tex]\[
A = 3x \times (-x)
\][/tex]
Now, let's calculate this product step-by-step:
1. Multiply the coefficients [tex]\(3\)[/tex] and [tex]\(-1\)[/tex]:
[tex]\[ 3 \times (-1) = -3 \][/tex]
2. Multiply the variable parts [tex]\(x\)[/tex] and [tex]\(x\)[/tex]:
[tex]\[ x \times x = x^2 \][/tex]
Combining these results, we get:
[tex]\[
A = -3 \times x^2 = -3x^2
\][/tex]
Therefore, the value of [tex]\( A \)[/tex] is:
[tex]\[
-3x^2
\][/tex]
So, the correct answer is [tex]\(\boxed{-3x^2}\)[/tex].
