Beginning Algebra With Applications, Chapter 8, 8.1, Section 8.1, Problem 130

Write an expression in factored form for the shaded portion in the diagram.

a.







The area of the rectangle is equal to $A_{\text{rectangle}} = LW$, where $L = 4r$ and $W = 2r$. So,

$A_{\text{rectangle}} = 4r(2r) = 8r^2$

And the area of the two circles is


$\begin{equation} \begin{aligned} A_{\text{circle}} =& 2 (\pi r^2) \\ =& 2 \pi r^2 \end{aligned} \end{equation}$


Then, by subtracting the area of the rectangle to the area of circle, we get the area of the shaded portion as


$\begin{equation} \begin{aligned} A_{\text{rectangle}} - A_{\text{circle}} =& 8r^2 - 2 \pi r^2 \\ =& 2r^2 (4 - \pi) \end{aligned} \end{equation}$


b.







Based from the figure, the area of the square is $A_{\text{rectangle}} = (2r)^2 = 4r^2$ and the area of the circle is $A_{\text{circle}} = \pi r^2$
Then, by subtracting the area of the square to the area of the circle, we obtain the area of the shaded portion as



$\begin{equation} \begin{aligned} A_{\text{rectangle}} - A_{\text{circle}} =& 4r^2 - \pi r^2 \\ =& r^2(4 - \pi) \end{aligned} \end{equation}$