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