Question

Enter your answer and show all the steps that you use to solve this problem in the space provided.

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

area of shaded region: x² + 23x + 49

solve:

Area of full rectangle:

`Length * Width`

`(x + 10 ) *( 2x + 5)`

`2x^2 + 5x + 20x + 50`

`2x^2 + 25x + 50`

Area of un-shaded rectangle:

`Length * Width`

`(x+1)(x+1)`

`x^2 + x+ x+ 1`

`x^2 + 2x + 1`

Area of shaded = Area of full rectangle - Area of un-shaded rectangle

`\hookrightarrow 2x^2 + 25x + 50 - (x^2 + 2x + 1)`

`\hookrightarrow 2x^2 + 25x + 50 - x^2 - 2x - 1`

`\hookrightarrow x^2 + 23x +49`

Answer 2

• x² + 23x + 49

Explanation:

The shaded area is the difference of areas of the rectangle and the square.

Rectangle:

• A₁ = (x + 10)(2x + 5) = 2x² + 5x + 20x + 50 = 2x² + 25x + 50

Square:

• A₂ = (x + 1)² = x² + 2x + 1

Shaded region:

• A₁ - A₂ =
• 2x² + 25x + 50 - (x² + 2x + 1) =
• 2x² + 25x + 50 - x² - 2x - 1 =
• x² + 23x + 49