Question

use the area model to find the product (-4x + 2)(-4x - 2). first, find the partial products. -4x -2 -4x -16x -8 2 -8x -4 now, write the product. (-4x + 2)(-4x - 2) =

Explanation:

Step1: Identify partial products

The partial products from the area model are \(-16x^2\), \(-8x\), \(-8x\), and \(-4\)? Wait, no, wait. Wait, the first row: \(-4x \times -4x = 16x^2\)? Wait, no, the yellow box: \(-4x \times -4x\) should be \(16x^2\), but the label is \(-16x\)? Wait, no, maybe I misread. Wait, the columns are \(-4x\) and \(-2\), rows are \(-4x\) and \(2\). So:

Yellow: \(-4x \times -4x = 16x^2\) (but the label is \(-16x\)? Wait, no, maybe the original problem has a typo? Wait, no, the user's image: yellow box is \(-16x\), teal is \(-8\), orange is \(-8x\), pink is \(-4\). Wait, maybe I misread the dimensions. Wait, the first row is \(-4x\) (row) and columns \(-4x\) and \(-2\). So first cell: \(-4x \times -4x = 16x^2\), but the label is \(-16x\)? That can't be. Wait, maybe the rows are \(-4x\) and \(2\), columns are \(-4x\) and \(-2\). So:

First row (y=-4x):
- Column x=-4x: \(-4x \times -4x = 16x^2\) (but label is \(-16x\)? No, maybe the label is wrong? Wait, no, the user's image: yellow box has \(-16x\), teal \(-8\), orange \(-8x\), pink \(-4\). Wait, maybe the rows are \(-4x\) and \(2\), columns \(-4x\) and \(-2\), so:

Wait, no, let's recalculate. The product \((-4x + 2)(-4x - 2)\) is a difference of squares: \((a + b)(a - b) = a^2 - b^2\), where \(a = -4x\), \(b = 2\). So the product should be \((-4x)^2 - (2)^2 = 16x^2 - 4\). But let's use the area model.

Wait, the partial products:

First cell (top left): \(-4x \times -4x = 16x^2\) (but the label is \(-16x\)? That's a mistake? Wait, no, maybe the rows are \(-4x\) and \(2\), columns \(-4x\) and \(-2\), so:

Top left: \(-4x \times -4x = 16x^2\)

Top right: \(-4x \times -2 = 8x\)

Bottom left: \(2 \times -4x = -8x\)

Bottom right: \(2 \times -2 = -4\)

But the labels are \(-16x\), \(-8\), \(-8x\), \(-4\). So maybe the rows are \(-4x\) and \(2\), columns \(-4x\) and \(-2\), but the first cell is \(-4x \times -4x = -16x\)? No, that's incorrect. Wait, maybe the original problem has a typo, but let's go with the given partial products.

Wait, the partial products are \(-16x^2\)? Wait, no, the labels: yellow is \(-16x\) (maybe \(-16x^2\)?), teal \(-8\), orange \(-8x\), pink \(-4\). Wait, maybe the first cell is \(-4x \times -4x = 16x^2\) but labeled as \(-16x\) (wrong), but let's check the user's image again. Wait, the user's image:

Rows: first row is \(-4x\) (left), second row is \(2\) (left).

Columns: first column \(-4x\) (top), second column \(-2\) (top).

So:

Yellow (row \(-4x\), column \(-4x\)): \(-4x \times -4x = 16x^2\) (but label is \(-16x\) – maybe a typo, should be \(16x^2\))

Teal (row \(-4x\), column \(-2\)): \(-4x \times -2 = 8x\) (label is \(-8\) – typo, should be \(8x\) or \(-8\)? Wait, \(-4x \times -2 = 8x\), but label is \(-8\) – no, that's a constant. Wait, maybe the columns are \(-4x\) and \(-2\), rows are \(-4x\) and \(2\), so:

Yellow: \(-4x \times -4x = 16x^2\)

Teal: \(-4x \times -2 = 8x\)

Orange: \(2 \times -4x = -8x\)

Pink: \(2 \times -2 = -4\)

Ah! Now I see. The labels in the image are wrong? Wait, no, the user's image: yellow is \(-16x\), teal \(-8\), orange \(-8x\), pink \(-4\). So maybe the rows are \(-4x\) and \(2\), columns \(-4x\) and \(-2\), but the first cell is \(-4x \times -4x = -16x\) (which is wrong, should be \(16x^2\)), teal is \(-4x \times -2 = -8\) (correct, because \(-4x \times -2 = 8x\)? No, \(-4x \times -2 = 8x\), but label is \(-8\) (constant). This is confusing. Wait, maybe the problem is \((-4x + 2)(-4x - 2)\), which is a difference of squares: \(a^2 - b^2\) where \(a = -4x\), \(b = 2\). So \((-4x)^2 - (2)^2 = 16x^2 - 4\). But let's use the partial products as per the image, even if labels are wrong. Wait, the partial products in the image are \(-16x\), \(-8\), \(-8x\), \(-4\)? No, that can't be. Wait, no, the yellow box is \(-16x^2\)? Maybe the label is \(-16x^2\) (missing the square). Let's assume that the partial products are \(16x^2\), \(8x\), \(-8x\), and \(-4\). Then combining like terms: \(16x^2 + (8x - 8x) - 4 = 16x^2 - 4\). But in the image, the labels are \(-16x\), \(-8\), \(-8x\), \(-4\). Wait, maybe the first cell is \(-4x \times -4x = -16x^2\) (wrong, should be positive), but let's proceed. Wait, no, the correct formula for \((a + b)(a - b)\) is \(a^2 - b^2\). Here, \(a = -4x\), \(b = 2\), so \((-4x)^2 - (2)^2 = 16x^2 - 4\). Alternatively, using the area model, sum the partial products:

From the image, the partial products are \(-16x^2\) (assuming yellow is \(16x^2\) but labeled \(-16x\) – no, maybe the labels are:

Yellow: \(-4x \times -4x = 16x^2\) (but label is \(-16x\) – maybe a typo, should be \(16x^2\))

Teal: \(-4x \times -2 = 8x\) (label \(-8\) – typo)

Orange: \(2 \times -4x = -8x\) (label \(-8x\) – correct)

Pink: \(2 \times -2 = -4\) (label \(-4\) – correct)

So sum the partial products: \(16x^2 + 8x - 8x - 4\). The \(8x\) and \(-8x\) cancel, leaving \(16x^2 - 4\).

Wait, but in the image, the yellow label is \(-16x\), teal \(-8\), orange \(-8x\), pink \(-4\). So if we take the labels as given (even if wrong), sum them: \(-16x - 8 - 8x - 4\) – that would be \(-24x -12\), which is wrong. So clearly, the labels in the image have typos. The correct partial products should be:

- \(-4x \times -4x = 16x^2\)

- \(-4x \times -2 = 8x\)

- \(2 \times -4x = -8x\)

- \(2 \times -2 = -4\)

Summing these: \(16x^2 + 8x - 8x - 4 = 16x^2 - 4\).

So the product is \(16x^2 - 4\).

Step2: Sum the partial products

Assuming the correct partial products (correcting the typos), we have \(16x^2 + 8x - 8x - 4\). The \(8x\) and \(-8x\) are like terms and cancel out, leaving \(16x^2 - 4\).

Answer:

\(16x^2 - 4\)