Write an expression for the area of the shaded region in its simplest form. Show all of your steps.

The whole shaded area is a rectangle that is:

x+10 by 2x+5

Then there is a plain white square inside the larger rectangle that is:

x+1 by x+1

The area of the shaded region is going to be the area of the rectangle minus the area of the square.

Area of a rectangle is L * W.

A = L * W

A = (x + 10) (2x + 5)

A = x (2x + 5) + 10 (2x + 5)

A = 2x^2 + 5x + 20x + 50

A = 2x^2 + 25x + 50 ... this is the area of the rectangle

area of a square is : A = a^2 ... where a is one side

A = (x + 1) ^2

A = (x + 1) (x + 1)

A = x (x + 1) + 1 (x + 1)

A = x^2 + x + x + 1

A = x^2 + 2x + 1

now we subtract the area of the square from the area of the rectangle to get the area of the shaded region.

2x^2 + 25x + 50 - (x^2 + 2x + 1) =

2x^2 + 25x + 50 - x^2 - 2x - 1 =

x^2 + 23x + 49 < = = the area of the shaded region