Mr. Randolph is putting the finishing touches on his backyard flower bed. The flower bed is $25$ feet long and $6$ feet wide, and it is bordered on two sides by the backyard fence. Mr. Randolph plans to put a pathway around the other two sides of the flower bed that is the same width on both sides. $\newline$ The total area of the flower bed and pathway in square feet can be modeled by the expression $(x + 25)(x + 6)$ , where $x$ is the width of the pathway in feet. This expression can also be written in the form $x^2 + 31x + 150$ . What does the quantity $x^2 + 31x$ represent in the expression? $\newline$ Choices: $\newline$ (A)the area of the flower bed in square feet $\newline$ (B)the area of the pathway in square feet $\newline$ (C)the perimeter around the outside of the flower bed and pathway in feet $\newline$ (D)the length of the flower bed and pathway together in feet

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Q. Mr. Randolph is putting the finishing touches on his backyard flower bed. The flower bed is

25

feet long and

6

feet wide, and it is bordered on two sides by the backyard fence. Mr. Randolph plans to put a pathway around the other two sides of the flower bed that is the same width on both sides.

\newline

The total area of the flower bed and pathway in square feet can be modeled by the expression

(x + 25)(x + 6)

, where

x

is the width of the pathway in feet. This expression can also be written in the form

x^2 + 31x + 150

. What does the quantity

x^2 + 31x

represent in the expression?

\newline

Choices:

\newline

(A)the area of the flower bed in square feet

\newline

(B)the area of the pathway in square feet

\newline

(C)the perimeter around the outside of the flower bed and pathway in feet

\newline

(D)the length of the flower bed and pathway together in feet

Total Area Expression: The expression for the total area is $(x + 25)(x + 6)$ , which expands to $x^2 + 31x + 150$ .
Area of Pathway Squared: The term $x^2$ represents the area of the pathway squared, since the pathway is added to both the length and width of the flower bed.
Area of Pathway along Length and Width: The term $31x$ is the sum of $25x$ and $6x$ , which represents the area of the pathway along the length of the flower bed ( $25x$ ) and the area of the pathway along the width of the flower bed ( $6x$ ).
Constant Term $150$ : The constant term $150$ represents the area of the flower bed itself, which is $25$ feet by $6$ feet.
Total Area of Pathway: Therefore, the quantity $x^2 + 31x$ represents the total area of the pathway around the flower bed.