Question

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

Explanation:

Step1: Analyze length of x², x, and 1 sections

Count the number of \( x^2 \) tiles: \( 3 \times 3 = 9 \)? Wait, no, looking at the grid: the blue \( x^2 \) tiles are 3 columns (wait, no, let's check rows and columns. Wait, the first part (blue \( x^2 \)): let's see the horizontal and vertical. Wait, the \( x^2 \) are in a 3x3? No, wait the green \( x \) and yellow \( 1 \). Wait, let's find the length of each side.

For the first factor (horizontal): Let's see the number of \( x^2 \), \( x \), and 1 in one row. Wait, the blue \( x^2 \) are 3 (columns), then green \( x \) are 3 (columns), then? Wait no, looking at the grid:

Wait, the top part: blue \( x^2 \) has 3 columns (since 3 \( x^2 \) in a row), then green \( x \) has 3 columns. Then vertically, the blue \( x^2 \) has 3 rows? Wait no, the blue \( x^2 \) tiles: let's count. The blue tiles: 3 columns (each \( x^2 \)) and 3 rows? Wait no, the blue tiles are 3 columns (each \( x^2 \)) and 3 rows? Wait, no, looking at the grid:

Wait, the first part (left of green \( x \)): the blue \( x^2 \) are 3 columns (each \( x^2 \)) and 3 rows? Wait, no, the blue \( x^2 \) tiles: 3 columns (so 3x) and 3 rows? Wait, no, let's check the green \( x \) tiles. The green \( x \) tiles: in the horizontal direction, after the blue \( x^2 \), there are 3 green \( x \) columns (each \( x \)) and then yellow \( 1 \) columns? Wait, no, the yellow \( 1 \) tiles: 3 columns (each 1) and 2 rows? Wait, no, let's look at the vertical and horizontal dimensions.

Wait, the vertical direction (rows): the blue \( x^2 \) has 3 rows? No, the blue \( x^2 \) tiles: 3 rows (each row has 3 \( x^2 \))? Wait, no, the first three rows (top three) have blue \( x^2 \) (3 in a row) and green \( x \) (3 in a row). Then the bottom two rows (rows 4 and 5) have green \( x \) (3 in a row) and yellow \( 1 \) (3 in a row). Wait, no, the grid:

Wait, rows: let's count the number of rows with \( x^2 \): 3 rows (top three). Then rows with \( x \) (green) below? Wait, no, the bottom two rows (rows 4 and 5) have green \( x \) (3 columns) and yellow \( 1 \) (3 columns). Wait, no, the vertical length (number of rows) for the first part (left of green \( x \)): the blue \( x^2 \) are in 3 rows, and the green \( x \) below (rows 4 and 5) are 2 rows? Wait, no, let's check the options.

Wait, the options are:

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

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

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

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

Wait, let's find the length of each side (the two factors). Let's look at the horizontal (first factor) and vertical (second factor).

Horizontal (first factor): Let's count the number of \( x \) and constant terms.

Looking at the horizontal direction (columns):

- The number of \( x^2 \) columns: 3 (so 3x)

- Then the number of \( x \) columns: 3 (so 3x? No, wait, the green \( x \) tiles: in the horizontal direction, after the blue \( x^2 \), there are 3 green \( x \) columns. Then, in the vertical direction (rows):

- The number of \( x^2 \) rows: 3 (so 3x)

- Then the number of \( x \) rows: 2 (so 2x? No, wait, the bottom two rows (rows 4 and 5) have green \( x \) and yellow \( 1 \). Wait, no, let's count the number of \( x^2 \), \( x \), and 1 in each dimension.

Wait, the blue \( x^2 \) tiles: 3 columns (so 3x) and 3 rows (so 3x)? No, that can't be. Wait, the blue \( x^2 \) are 3 columns (each \( x^2 \)) and 3 rows? Wait, no, the blue \( x^2 \) tiles: 3 columns (so 3x) and 3 rows (so 3x)? Then the green \( x \) tiles: 3 columns (so 3x) and 2 rows (so 2x)? No, this is confusing. Wait, let's use the options.

Let's expand each option:

Option 1: \( (3x + 2)(2x + 2) = 6x^2 + 6x + 4x + 4 = 6x^2 + 10x + 4 \)

Option 2: \( (3x + 3)(3x + 2) = 9x^2 + 6x + 9x + 6 = 9x^2 + 15x + 6 \)

Option 3: \( (3x + 3)(2x + 2) = 6x^2 + 6x + 6x + 6 = 6x^2 + 12x + 6 \)

Option 4: \( (3x + 2)(3x + 2) = 9x^2 + 6x + 6x + 4 = 9x^2 + 12x + 4 \)

Now, let's count the number of \( x^2 \), \( x \), and 1 tiles in the grid.

- \( x^2 \) tiles: 3 columns (each \( x^2 \)) and 3 rows? Wait, no, the blue \( x^2 \) tiles: looking at the grid, the blue \( x^2 \) are in a 3x3 grid? Wait, no, the first three rows (top three) have blue \( x^2 \) (3 in a row) and green \( x \) (3 in a row). Then the bottom two rows (rows 4 and 5) have green \( x \) (3 in a row) and yellow \( 1 \) (3 in a row). Wait, no, the blue \( x^2 \) tiles: 3 columns (each \( x^2 \)) and 3 rows (so 3*3=9 \( x^2 \) tiles? But that would be 9 \( x^2 \), but let's check the options. Wait, no, the blue \( x^2 \) tiles: 3 columns (so 3x) and 3 rows (so 3x)? Then the green \( x \) tiles: 3 columns (so 3x) and 2 rows (so 2x)? No, this is wrong. Wait, let's look at the horizontal and vertical lengths.

Wait, the first factor (horizontal) should be the sum of the lengths of \( x^2 \), \( x \), and 1 in one row. Wait, in one row (top row), we have 3 \( x^2 \) (so 3x) and 3 \( x \) (so 3x)? No, that can't be. Wait, no, the blue \( x^2 \) are 3 columns (so 3x) and the green \( x \) are 3 columns (so 3x)? No, that would be 3x + 3x, but no. Wait, maybe the horizontal length is (3x + 3) and vertical length is (3x + 2)? No, let's count the number of \( x^2 \), \( x \), and 1.

Wait, the \( x^2 \) tiles: 3 columns (each \( x^2 \)) and 3 rows? No, the blue \( x^2 \) tiles: 3 columns (so 3x) and 3 rows (so 3x)? Then the green \( x \) tiles: 3 columns (so 3x) and 2 rows (so 2x)? No, this is not working. Wait, let's count the number of \( x^2 \) tiles: 3 columns (each \( x^2 \)) and 3 rows? Wait, the blue \( x^2 \) tiles: 3 columns (so 3x) and 3 rows (so 3x) → 9 \( x^2 \) tiles. Then the green \( x \) tiles: 3 columns (so 3x) and 2 rows (so 2x) → 3*2=6 \( x \) tiles? No, the green \( x \) tiles: in the top 3 rows, 3 columns of \( x \) (so 3*3=9 \( x \) tiles), and in the bottom 2 rows, 3 columns of \( x \) (so 2*3=6 \( x \) tiles) → total \( x \) tiles: 9 + 6 = 15. Then the yellow \( 1 \) tiles: 3 columns (so 3*1) and 2 rows (so 2*1) → 3*2=6 \( 1 \) tiles. Now, let's check the expansion of \( (3x + 3)(3x + 2) \):

\( (3x + 3)(3x + 2) = 9x^2 + 6x + 9x + 6 = 9x^2 + 15x + 6 \). Which matches the count: 9 \( x^2 \), 15 \( x \), 6 \( 1 \). Let's verify:

- \( x^2 \) tiles: 3x * 3x = 9x² → correct (9 tiles)
- \( x \) tiles: 3x * 2 + 3 * 3x = 6x + 9x = 15x → correct (15 tiles)
- \( 1 \) tiles: 3 * 2 = 6 → correct (6 tiles)

Now check other options:

Option 1: \( (3x + 2)(2x + 2) = 6x² + 10x + 4 \). \( x² \) would be 6, but we have 9 \( x² \) tiles → wrong.

Option 3: \( (3x + 3)(2x + 2) = 6x² + 12x + 6 \). \( x² \) is 6, but we have 9 → wrong.

Option 4: \( (3x + 2)(3x + 2) = 9x² + 12x + 4 \). \( 1 \) tiles would be 4, but we have 6 → wrong.

So the correct expression is \( (3x + 3)(3x + 2) \).

Step2: Confirm with expansion

Expand \( (3x + 3)(3x + 2) \):

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

\( = 9x² + 6x + 9x + 6 \)

\( = 9x² + 15x + 6 \)

Which matches the tile count (9 \( x² \), 15 \( x \), 6 \( 1 \)).

Answer:

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