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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.
To find the area of the shaded region, we first need to find the area of the entire rectangle and then subtract the area of the unshaded square.
Area of rectangle = length * width
= (x + 10)(2x + 5)
= 2x^2 + 5x + 20x + 50
= 2x^2 + 25x + 50
Area of unshaded square = (x + 1)(x + 1)
= x^2 + x + x + 1
= x^2 + 2x + 1
Area of shaded region = Area of rectangle - Area of unshaded square
= 2x^2 + 25x + 50 - (x^2 + 2x + 1)
= 2x^2 + 25x + 50 - x^2 - 2x - 1
= x^2 + 23x + 49
Therefore, the expression for the area of the shaded region in its simplest form is x^2 + 23x + 49.