Question

1. in the diagram at the right, the dimensions of the large rectangle are (3x - 1) by (3x + 7) units. the dimensions of the cut - out rectangle are x by 2x + 5 units. which choice expresses the area of the shaded region, in square units? 1 $x^{2}+23x - 7$ 2 $x^{2}+13x - 7$ 3 $7x^{2}+23x - 7$ 4 $7x^{2}+13x - 7$

Explanation:

Step1: Find area of large rectangle

The area formula for a rectangle is $A = l\times w$. For the large rectangle with length $l=(3x + 7)$ and width $w=(3x - 1)$, we use the FOIL method.
\[

$$\begin{align*} A_{1}&=(3x - 1)(3x + 7)\\ &=3x\times3x+3x\times7-1\times3x - 1\times7\\ &=9x^{2}+21x-3x - 7\\ &=9x^{2}+18x - 7 \end{align*}$$

\]

Step2: Find area of cut - out rectangle

For the cut - out rectangle with length $l = 2x+5$ and width $w=x$, the area is $A_{2}=x(2x + 5)=2x^{2}+5x$.

Step3: Find area of shaded region

The area of the shaded region $A$ is the area of the large rectangle minus the area of the cut - out rectangle.
\[

$$\begin{align*} A&=A_{1}-A_{2}\\ &=(9x^{2}+18x - 7)-(2x^{2}+5x)\\ &=9x^{2}+18x - 7-2x^{2}-5x\\ &=(9x^{2}-2x^{2})+(18x - 5x)-7\\ &=7x^{2}+13x - 7 \end{align*}$$

\]

Answer:

D. $7x^{2}+13x - 7$