Question

You want to build a rectangular garden of a certain size in your backyard. The dimensions of your garden are represented by the picture. (The length is 3x + 7, and the width is 2x - 4.) Part A: write an expression that represents the area of the garden.

Part B: use the area model to find the area of the garden

Part C: the simplified polynomial written in standard form is

Answer 1

To solve this problem, we'll work through each part step by step.

### Part A: Write an expression that represents the area of the garden.

The area \( A \) of a rectangle is calculated with the formula:

\[
A = \text{length} \times \text{width}
\]

Given the length \( L = 3x + 7 \) and the width \( W = 2x - 4 \), the expression for the area of the garden is:

\[
A = (3x + 7)(2x - 4)
\]

### Part B: Use the area model to find the area of the garden.

Now we will use the distributive property (also known as the area model) to expand the expression from Part A:

\[
A = (3x + 7)(2x - 4)
\]

We can expand this by multiplying each term in the first parentheses by each term in the second parentheses:

1. \( 3x \times 2x = 6x^2 \)
2. \( 3x \times (-4) = -12x \)
3. \( 7 \times 2x = 14x \)
4. \( 7 \times (-4) = -28 \)

Now we combine all these results:

\[
A = 6x^2 - 12x + 14x - 28
\]

Next, combine like terms (\(-12x\) and \(14x\)):

\[
A = 6x^2 + 2x - 28
\]

### Part C: The simplified polynomial written in standard form is.

The standard form of a polynomial is written with the terms in descending order of degree.

Thus, the final expression for the area of the garden, simplified and written in standard form, is:

\[
A = 6x^2 + 2x - 28
\]

To summarize:
- Part A: \( A = (3x + 7)(2x - 4) \)
- Part B: \( A = 6x^2 + 2x - 28 \)
- Part C: \( 6x^2 + 2x - 28 \)