Question

the model represents a polynomial and its factors. which equation is represented by the model? $x^2 - 2x - 3 = (x - 3)(x + 1)$ $x^2 - 4x + 3 = (x - 3)(x - 1)$ $x^2 + 2x + 3 = (x + 3)(x - 1)$ $x^2 + 4x - 3 = (x + 3)(x + 1)$

Explanation:

Step1: Analyze the polynomial from the model

First, we identify the terms in the polynomial. The orange region has \(+x^2\), \(+x\) (top row), \(+x\) (left column), and the blue region has \(-x\) (three times), \(-x\) (middle row left), \(-\) (top row three times), and \(+\) (bottom row three times). Wait, maybe a better way is to calculate the polynomial by adding all the terms. The big square: \(x^2\), then the \(x\) terms: \(+x + x - x - x - x - x\)? Wait, no, let's look at the grid. The rows and columns: let's see the length and width of the rectangle. The length seems to be \(x - 3\) and width \(x - 1\)? Wait, no, let's expand each option.

Step2: Expand each option

• Option 1: \((x - 3)(x + 1)=x^2 + x - 3x - 3 = x^2 - 2x - 3\). But let's check the model.
• Option 2: \((x - 3)(x - 1)=x^2 - x - 3x + 3 = x^2 - 4x + 3\). Let's see the model: the polynomial has \(x^2\), then \(x\) terms: let's count the \(x\) tiles. The orange has \(x^2\), \(+x\) (top), \(+x\) (left), and the blue has \(-x\) (three times in middle), \(-x\) (middle left), \(-\) (top three), and \(+\) (bottom three). Wait, maybe the polynomial is \(x^2 - 4x + 3\). Let's check the expansion of option 2: \(x^2 - 4x + 3\), which matches the expansion of \((x - 3)(x - 1)\). Let's verify the other options:
• Option 3: \((x + 3)(x - 1)=x^2 - x + 3x - 3 = x^2 + 2x - 3\), but the option says \(x^2 + 2x + 3\), so wrong.
• Option 4: \((x + 3)(x + 1)=x^2 + x + 3x + 3 = x^2 + 4x + 3\), but the option says \(x^2 + 4x - 3\), wrong.

Wait, maybe I made a mistake in the first analysis. Let's re - expand option 2: \((x - 3)(x - 1)=x^2 - x - 3x + 3 = x^2 - 4x + 3\). Now, looking at the model, the polynomial has \(x^2\), then the linear terms: let's see the number of \(x\) terms. If we have a rectangle with length \(x - 3\) and width \(x - 1\), that would make sense. So the correct expansion is option 2.

Answer:

B. \(x^2 - 4x + 3=(x - 3)(x - 1)\) (assuming the options are labeled as A, B, C, D with B being the second option: \(x^2 - 4x + 3=(x - 3)(x - 1)\))