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A rectangle is shown with length x plus 10 and width 2 x plus 5. The inside of the rectangle is shaded other than an unshaded square with length x plus 1 and width x plus 1.
Write an expression for the area of the shaded region in its simplest form. Show all of your steps.
(2 points)
To find the area of the shaded region, we need to subtract the area of the unshaded square from the area of the rectangle.
The area of the rectangle is given by the formula: length × width.
Length of the rectangle = x + 10
Width of the rectangle = 2x + 5
The area of the rectangle = (x + 10) * (2x + 5)
The area of the unshaded square is given by the formula: side × side.
Side of the square = x + 1
The area of the unshaded square = (x + 1) * (x + 1)
Now, we subtract the area of the unshaded square from the area of the rectangle to find the area of the shaded region.
Area of the shaded region = (x + 10) * (2x + 5) - (x + 1) * (x + 1)
To simplify the expression, we can use the distributive property to expand the terms:
Area of the shaded region = (2x^2 + 5x + 20x + 50) - (x^2 + 2x + 1)
Now, combine like terms:
Area of the shaded region = 2x^2 + 25x + 50 - x^2 - 2x - 1
Simplify further:
Area of the shaded region = x^2 + 23x + 49
Therefore, the expression for the area of the shaded region is x^2 + 23x + 49.