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.

Answer 1

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.