Question

find the area of the rectangle. a 9x + 3 b 9x^2+5x + 2 c 14x^2+2 d 14x^2+11x + 2

Explanation:

Step1: Find the length of the rectangle

The length of the rectangle is the sum of the parts along the length - side, which is $(2x + 1)+2=2x + 3$.

Step2: Find the width of the rectangle

The width of the rectangle is $7x$.

Step3: Use the area formula for a rectangle

The area formula of a rectangle is $A = l\times w$, where $l$ is the length and $w$ is the width. Substitute $l = 2x + 3$ and $w=7x$ into the formula:
\[

$$\begin{align*} A&=(2x + 3)\times7x\\ &=2x\times7x+3\times7x\\ &=14x^{2}+21x \end{align*}$$

\]
There seems to be an error in the provided options as the correct area is $14x^{2}+21x$. If we assume there is a mis - take in the problem setup and we consider the length as $2x + 1$ and width as $7x$ (ignoring the extra part of length calculation above), then:
\[

$$\begin{align*} A&=(2x + 1)\times7x\\ &=2x\times7x+1\times7x\\ &=14x^{2}+7x \end{align*}$$

\]
Still, this does not match the options. If we assume the length is $2x+1$ and we made a wrong start above, and we calculate the area as follows:

Step1: Identify length and width

Length $l = 2x + 1$, width $w = 7x$.

Step2: Calculate area

\[

$$\begin{align*} A&=l\times w\\ &=(2x + 1)\times7x\\ &=2x\times7x+7x\times1\\ &=14x^{2}+7x \end{align*}$$

\]
If we assume the length is $2x + 1$ and we calculate the area using the distributive property $A=(2x + 1)\times7x=14x^{2}+7x$. But if we consider the correct way of adding up the length components:

Step1: Calculate length

Length of rectangle $L=(2x + 1)+2=2x+3$.
Width of rectangle $W = 7x$.

Step2: Use area formula

\[

$$\begin{align*} A&=L\times W\\ &=(2x + 3)\times7x\\ &=2x\times7x+3\times7x\\ &=14x^{2}+21x \end{align*}$$

\]
If we assume the length is just $2x + 1$ (simplified view ignoring the extra part in the figure that might be a mis - draw or mis - understanding)

Step1: Recall area formula

Area of rectangle $A = lw$, where $l = 2x+1$ and $w = 7x$.

Step2: Expand the product

\[

$$\begin{align*} A&=(2x + 1)\times7x\\ &=2x\times7x+1\times7x\\ &=14x^{2}+7x \end{align*}$$

\]
If we assume the length is $2x+1$ and width is $7x$:

Answer:

None of the given options (A. $9x + 3$, B. $9x^{2}+5x + 2$, C. $14x^{2}+2$, D. $14x^{2}+11x + 2$) are correct. The correct area if length $l = 2x + 1$ and width $w=7x$ is $14x^{2}+7x$, and if length $l=2x + 3$ and width $w = 7x$ is $14x^{2}+21x$.