which model represents the factors of 4x² - 9?

Explanation:

Step1: Recognize difference - of - squares formula

The expression $4x^{2}-9$ is in the form $a^{2}-b^{2}$, where $a = 2x$ ($(2x)^{2}=4x^{2}$) and $b = 3$ ($3^{2}=9$). The difference - of - squares formula is $a^{2}-b^{2}=(a + b)(a - b)$. So, $4x^{2}-9=(2x + 3)(2x-3)$.

Step2: Analyze the area - model representation

We need to find the model that represents the product $(2x + 3)(2x-3)$. When we expand $(2x + 3)(2x-3)$ using the FOIL method: $(2x)\times(2x)+(2x)\times(-3)+3\times(2x)+3\times(-3)=4x^{2}-6x + 6x-9=4x^{2}-9$. The first term $4x^{2}$ comes from the product of the first terms of the binomials $(2x\times2x)$, the outer and inner terms $-6x$ and $6x$ cancel each other out, and the last term is $-9$. We look for a model with four $x^{2}$ tiles (to represent $4x^{2}$), and a combination of positive and negative $x$ - tiles that cancel each other out, and nine negative unit tiles (to represent $-9$).

Answer:

The model that has four $+x^{2}$ tiles, some positive and negative $x$ - tiles that cancel each other out, and nine negative unit tiles represents the factors of $4x^{2}-9$. Without seeing all the options clearly, the correct model should be the one that can be factored into $(2x + 3)(2x-3)$ based on the area - model concept where the product of the binomials gives the polynomial $4x^{2}-9$. If we assume the first model has the correct combination of tiles as described above, it is the correct one.