Question

which multiplication expression does this area model represent?
(2x + 2)(2x + 3) (x + 2)(2x + 3)
(2x + 3)(2x + 3) (x + 3)(2x + 3)

Explanation:

Step1: Analyze the length of the rectangle (horizontal side)

Looking at the area model, the horizontal side has terms: \(x^2\) (two of them, so \(2x^2\) when combined? Wait, no, let's count the number of \(x\) and constant terms. Wait, actually, let's look at the columns. The first part (blue and green \(x\)s) and the second part (green \(x\)s and yellow 1s). Wait, maybe better to count the number of \(x\) terms and constant terms in each dimension.

Wait, the vertical dimension (height) and horizontal dimension (width). Let's look at the vertical stacks:

First, the blue \(x^2\)s: there are 2 rows of \(x^2\), then 3 rows of \(x\). Wait, no, let's count the number of \(x\) terms in each dimension.

Wait, the horizontal direction (width) has: from the blue \(x^2\)s, there are 2 \(x^2\)s (so that's \(2x\) in terms of the length? Wait, no, maybe the length is composed of \(x + x + 1 + 1 + 1\)? No, let's look at the area model as a rectangle with length and width.

Wait, the area model can be divided into two parts for the length and two parts for the width? Wait, no, let's count the number of \(x\) terms and constant terms in each side.

Looking at the vertical direction (height):

- The top two rows have \(x^2\) and \(x\) terms. Wait, the first two rows (blue \(x^2\) and green \(x\)): each row has 1 \(x^2\) and 3 \(x\)s? No, the blue \(x^2\) is a square, then three \(x\)s (green) in the first row. Second row same: \(x^2\) and three \(x\)s. Then the next three rows: each has an \(x\) (green) and three 1s (yellow).

Wait, maybe the length (horizontal) is \(2x + 3\)? Wait, no, let's count the number of \(x\) terms and constant terms in the length and width.

Wait, the width (horizontal) has:

- Number of \(x\) terms: Let's see, the first part (left) has \(x^2\) (two of them, so that's \(2x\) when considering the length? Wait, no, maybe the length is \(x + x + 3\)? No, let's think of the area model as a rectangle with length \( (2x + 3) \) and width \( (x + 3) \)? No, wait, let's count the number of \(x\) terms and constant terms in each dimension.

Wait, the vertical dimension (height) has:

- The first two rows: each has \(x^2\) (so that's \(2x\) in terms of the height? Wait, no, \(x^2\) is a square with side \(x\), so two \(x^2\)s stacked vertically would be height \(2x\)? Wait, maybe the height is \(2x + 3\)? No, let's look at the answer choices.

Wait, the answer choices are:

1. \( (2x + 2)(2x + 3) \)

2. \( (x + 2)(2x + 3) \)

3. \( (2x + 3)(2x + 3) \)

4. \( (x + 3)(2x + 3) \)

Wait, let's calculate the area of the model. Let's count the number of \(x^2\), \(x\), and constant terms.

- \(x^2\) terms: 2 (blue squares)

- \(x\) terms: Let's see, the green \(x\)s: first two rows have 3 \(x\)s each, so \(2 \times 3 = 6\) \(x\)s. Then the next three rows have 1 \(x\) each, so \(3 \times 1 = 3\) \(x\)s. Total \(x\) terms: \(6 + 3 = 9\) \(x\)s? Wait, no, that can't be. Wait, maybe I'm miscounting.

Wait, the area model is a rectangle. Let's find the length and width by looking at the number of \(x\) terms and constant terms in each side.

Looking at the horizontal direction (width):

- The first part (left) has \(x^2\) (two of them, so that's \(2x\) in length? Wait, no, \(x^2\) is a square with side \(x\), so two \(x^2\)s side by side would be length \(2x\)? Wait, no, \(x^2\) is area, but in the area model, the length is composed of \(x\) terms and constant terms.

Wait, maybe the length (horizontal) is \( (x + x + 1 + 1 + 1) \)? No, that's \(2x + 3\). Wait, the width (vertical) is \( (x + x + 1 + 1 + 1) \)? No, let's look at the answer choices.

Wait, let's expand each option and see which one matches the area.

First, let's find the area of the model.

Count the number of \(x^2\) terms: 2 (blue squares: two \(x^2\)s).

Number of \(x\) terms: Let's see, the green \(x\)s:

- In the first two rows (top two rows), each row has 3 \(x\)s (green) and 1 \(x^2\) (blue). Wait, no, the blue \(x^2\) is a square, then three \(x\)s (green) in the first row. Second row: \(x^2\) and three \(x\)s. So that's 2 \(x^2\)s and 6 \(x\)s (3 per row, 2 rows).

Then the next three rows (bottom three rows): each row has 1 \(x\) (green) and three 1s (yellow). So that's 3 \(x\)s and 9 1s (3 per row, 3 rows).

So total \(x^2\) terms: 2.

Total \(x\) terms: 6 (from top two rows) + 3 (from bottom three rows) = 9.

Total constant terms: 9 (from bottom three rows, 3 per row, 3 rows).

Now let's expand each option:

1. \( (2x + 2)(2x + 3) \)

Expand: \(2x \times 2x + 2x \times 3 + 2 \times 2x + 2 \times 3 = 4x^2 + 6x + 4x + 6 = 4x^2 + 10x + 6\). But our model has 2 \(x^2\) terms, so this is too many \(x^2\)s. So eliminate.

2. \( (x + 2)(2x + 3) \)

Expand: \(x \times 2x + x \times 3 + 2 \times 2x + 2 \times 3 = 2x^2 + 3x + 4x + 6 = 2x^2 + 7x + 6\). \(x^2\) terms: 2, \(x\) terms: 7, constants: 6. But our model has 9 \(x\) terms and 9 constants. So no.

3. \( (2x + 3)(2x + 3) \)

Expand: \(2x \times 2x + 2x \times 3 + 3 \times 2x + 3 \times 3 = 4x^2 + 6x + 6x + 9 = 4x^2 + 12x + 9\). \(x^2\) terms: 4, which is more than our model's 2. Eliminate.

4. \( (x + 3)(2x + 3) \)

Expand: \(x \times 2x + x \times 3 + 3 \times 2x + 3 \times 3 = 2x^2 + 3x + 6x + 9 = 2x^2 + 9x + 9\).

Now let's check the model:

- \(x^2\) terms: 2 (matches).

- \(x\) terms: 3x + 6x = 9x (matches the 9 \(x\)s we counted).

- Constant terms: 9 (matches the 9 1s).

So this must be the correct expansion.

Wait, but let's re-examine the area model. Maybe I miscounted the \(x^2\) terms. Wait, the blue \(x^2\)s: there are two \(x^2\)s (two blue squares), so that's \(2x^2\) when combined? No, each \(x^2\) is a square with side \(x\), so two \(x^2\)s would be \(2x^2\) in area, but in terms of the length, maybe the length is \(x + 3\) and the width is \(2x + 3\)? Wait, no, the expansion of \( (x + 3)(2x + 3) \) is \(2x^2 + 9x + 9\), which matches the count: 2 \(x^2\), 9 \(x\), 9 constants. So that's correct.

Step2: Verify by counting the number of each term in the area model

- \(x^2\) terms: 2 (two blue squares: \(x^2 + x^2 = 2x^2\)).

- \(x\) terms:

- In the first two rows (top two rows), each row has 3 \(x\)s (green) and 1 \(x^2\). Wait, no, the first row: \(x^2\) (blue) + 3 \(x\)s (green) = \(x^2 + 3x\). Second row: same, \(x^2 + 3x\). So that's \(2x^2 + 6x\).

- Then the next three rows (bottom three rows), each row has 1 \(x\) (green) + 3 1s (yellow) = \(x + 3\). So three rows: \(3x + 9\).

- Total area: \( (x^2 + 3x) \times 2 + (x + 3) \times 3 \)? No, that's not the right way. Wait, no, the area model is a single rectangle, not two parts. So the length is \(x + 3\) and the width is \(2x + 3\), because when you multiply \( (x + 3)(2x + 3) \), you get \(2x^2 + 3x + 6x + 9 = 2x^2 + 9x + 9\), which matches the count of 2 \(x^2\), 9 \(x\), and 9 constants.

Answer:

\((x + 3)(2x + 3)\)