which area model best represents the product of the following factors?
$(x + 3)(2x - 5)$

Question

Accepted Answer

# Explanation:
## Step1: Expand the expression
To determine the area model, first expand $(x + 3)(2x - 5)$ using the distributive property (FOIL method).
$$
\begin{align*}
(x + 3)(2x - 5)&=x(2x - 5)+3(2x - 5)\
&=2x^2-5x + 6x-15\
&=2x^2+x - 15
\end{align*}
$$
## Step2: Analyze the area model components
An area model for $(a + b)(c + d)$ is a rectangle with length $(a + b)$ and width $(c + d)$, divided into four sub - rectangles with areas $ac$, $ad$, $bc$, and $bd$. For $(x + 3)(2x - 5)$, the length can be considered as $(2x-5)$ (which has terms $2x$ and $- 5$) and the width as $(x + 3)$ (which has terms $x$ and $3$) or vice - versa.
- The term $2x^2$ comes from $x\times2x$.
- The term $-5x$ comes from $x\times(-5)$.
- The term $6x$ comes from $3\times2x$.
- The term $-15$ comes from $3\times(-5)$.

Looking at the area models, we need to find the one where:
    - The number of $x^2$ terms is $2$ (since the coefficient of $x^2$ is $2$).
    - The number of $x$ terms: we have $-5x+6x = x$, so the net number of $x$ terms should be such that the positive $x$ terms (from $3\times2x$) and negative $x$ terms (from $x\times(-5)$) are represented correctly.
    - The constant term is $- 15$ (from $3\times(-5)$).

The correct area model should have:
    - Two regions with area $x^2$ (for the $2x^2$ term, since $x\times2x$ can be thought of as two rectangles each with area $x^2$ if we consider the $2x$ as two $x$ terms).
    - For the linear term: we have $-5x$ (from $x\times(-5)$) and $6x$ (from $3\times2x$). The $-5x$ would be represented as 5 regions with area $-x$ and $6x$ would be represented as 6 regions with area $x$, and the net is $x$ (6x - 5x=x).
    - The constant term $-15$ would be represented as 15 regions with area $-1$ (since $3\times(-5)=-15$).

After analyzing the given area models (by looking at the number of $x^2$ blocks, $x$ blocks, and constant blocks), the correct area model is the one where:
    - The part corresponding to $2x^2$ has two $x^2$ blocks.
    - The part corresponding to the $x$ terms has a net of one $x$ (after combining positive and negative $x$ terms).
    - The constant term is $-15$.

Among the given options, the fourth option (the one at the bottom right) has:
    - Two $x$ terms in the length (for the $2x$ part) and one $x$ term in the width (for the $x$ part) to get $2x^2$ (by multiplying the $x$ in the width with the two $x$s in the length).
    - The positive $x$ terms (from the $3$ in the width and $2x$ in the length) and negative $x$ terms (from the $x$ in the width and $-5$ in the length) are represented correctly to give a net of $x$.
    - The constant term (from $3$ in the width and $-5$ in the length) is $-15$.

# Answer: The bottom - right area model (the fourth one) best represents the product $(x + 3)(2x - 5)$.

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QUESTION IMAGE

Question

Explanation:

Step1: Expand the expression

Step2: Analyze the area model components

Answer: