Question

in the following figure, all angles formed by adjacent sides are right angles. which algebraic expression represents the area of the figure?\
\\(23x^2 + 28x\\)\
\\(22x^2 + 24x\\)\
\\(20x + 16\\)\
\\(18x^2 + 24x\\)

Explanation:

Step1: Decompose the figure

The figure can be seen as a combination of rectangles. Let's split it into parts. The vertical side on the right is \(6x + 8\), and we can find the horizontal lengths. The top and bottom horizontal segments: let's consider the total horizontal length. Also, the middle part and the side parts. Alternatively, we can use the method of subtracting or adding areas. Another way: the figure can be thought of as a large rectangle with some adjustments, but maybe easier to split into two rectangles. Wait, looking at the markings: the left side has \(x + 4\) and \(x\), and the top horizontal is \(3x\). Wait, maybe better to calculate the area by splitting the figure into two rectangles. Let's see: one rectangle with length \(3x\) and height \(x + 4\), and another rectangle? Wait, no, maybe the total height is \(6x + 8\), and the horizontal length: let's see the horizontal segments. Wait, maybe the figure is a composite of two rectangles. Wait, the right side is \(6x + 8\), and the horizontal length: let's find the length of the base. Wait, maybe the figure can be divided into two rectangles: one with dimensions \(3x\) and \(6x + 8\), and another? No, maybe not. Wait, looking at the left side: there are two segments, \(x + 4\) and \(x\), so total height is \((x + 4)+x=2x + 4\)? Wait, no, the right side is \(6x + 8\), which is equal to \(2(3x + 4)\). Wait, maybe the figure is a rectangle with length \(3x + 6\)? No, let's try to calculate the area by splitting into two parts. Let's consider the top rectangle: length \(3x\), height \(x + 4\), and the bottom rectangle: length \(3x\), height \(x + 4\)? No, that doesn't make sense. Wait, maybe the figure is a combination of a rectangle with length \(6x\) and height \(3x\), and another rectangle with length \(8\) and height \(3x\), and then another part? Wait, no, let's look at the answer choices. The answer choices have \(18x^2+24x\), \(22x^2+24x\), etc. Let's try to expand the area. Wait, maybe the figure is a rectangle with length \(3x + 6\) and height \(6x + 8\)? No, that's too complicated. Wait, let's use the method of adding areas. Let's split the figure into two rectangles: one with length \(3x\) and height \(6x + 8\), and another? No, maybe the figure is a composite of two rectangles: one with dimensions \(3x\) and \(x + 4\), and another with dimensions \(3x\) and \(x + 4\), and then a rectangle with length \(6\) and height \(3x\)? No, this is confusing. Wait, maybe the correct approach is to calculate the area as the sum of two rectangles. Let's see: the figure can be divided into two rectangles. The first rectangle has length \(3x\) and height \(x + 4\), and the second rectangle has length \(3x\) and height \(x + 4\), and then a rectangle with length \(6\) and height \(3x\)? No, that's not right. Wait, let's look at the answer choices. The answer is likely \(18x^2 + 24x\) or \(22x^2 + 24x\). Wait, let's try to calculate the area. Let's consider the figure as a rectangle with length \(3x\) and height \(6x + 8\), but that would be \(3x(6x + 8)=18x^2 + 24x\), but that's one of the options. Wait, but maybe there's another part. Wait, no, maybe the figure is a rectangle with length \(3x\) and height \(6x + 8\), but that would be \(3x(6x + 8)=18x^2 + 24x\), which is option D? Wait, no, the options are: 23x²+28x, 22x²+24x, 20x+16, 18x²+24x. Wait, let's check: \(3x(6x + 8)=18x^2 + 24x\), which is option D (18x²+24x). Wait, but let's verify. If the figure is a rectangle with length \(3x\) and height \(6x + 8\), then area is \(3x*(6x + 8)=18x^2 + 24x\), which matches option D. So that must be the area.

Step2: Calculate the area

The area of a rectangle is length times width. If we consider the figure as a rectangle with length \(3x\) and height \(6x + 8\), then:
\[ \begin{align*} \text{Area}&=3x\times(6x + 8)\\ &=3x\times6x+3x\times8\\ &=18x^2 + 24x \end{align*} \]

Answer:

\(18x^2 + 24x\) (the option corresponding to this expression, which is the last option: \(18x^2 + 24x\))