To express the area of the entire rectangle, we need to start by defining the dimensions of the rectangle. According to the given problem, both the length and the width of the rectangle are [tex]\( x + 3 \)[/tex].
Here are the steps to determine the area of the rectangle:
1. Identify the Length and Width:
- Length of the rectangle = [tex]\( x + 3 \)[/tex]
- Width of the rectangle = [tex]\( x + 3 \)[/tex]
2. Formula for Area of a Rectangle:
The area [tex]\( A \)[/tex] of a rectangle is given by multiplying its length [tex]\( L \)[/tex] by its width [tex]\( W \)[/tex].
[tex]\[
A = L \times W
\][/tex]
3. Substitute the Length and Width:
Here, [tex]\( L = x + 3 \)[/tex] and [tex]\( W = x + 3 \)[/tex]. So,
[tex]\[
A = (x + 3) \times (x + 3)
\][/tex]
4. Expand the Expression:
To find the standard form of the polynomial, we need to expand the expression [tex]\((x + 3)(x + 3)\)[/tex].
[tex]\[
(x + 3)(x + 3) = x \cdot x + x \cdot 3 + 3 \cdot x + 3 \cdot 3
\][/tex]
5. Perform the Multiplications:
Evaluate each of the products:
[tex]\[
x \cdot x = x^2
\][/tex]
[tex]\[
x \cdot 3 = 3x
\][/tex]
[tex]\[
3 \cdot x = 3x
\][/tex]
[tex]\[
3 \cdot 3 = 9
\][/tex]
6. Combine Like Terms:
Add the results of these multiplications together:
[tex]\[
x^2 + 3x + 3x + 9
\][/tex]
Combine the like terms (the [tex]\(3x\)[/tex] and [tex]\(3x\)[/tex]):
[tex]\[
x^2 + 6x + 9
\][/tex]
Therefore, the area of the rectangle, expressed as a polynomial in standard form, is:
[tex]\[
\boxed{x^2 + 6x + 9}
\][/tex]
