Chapter 4.5, Problem 42WE

To determine

To find: the area of the given shaded region

Answer to Problem 42WE

The area of the given shaded region is : $\boxed{2x^2+8x+32}$

Explanation of Solution

Given information: The shaded region is given below

  

Concept Involved:

Draw line in the composite figure so that it looks like combination of region with recognizable shapes.

Identify the area of each region and sum them together to find the area of composite shaded region.

Formula Used:

Area of rectangle = $Length\times Width$

Calculation:

Draw line in the composite figure so that it looks like combination of region with recognizable shapes.

  

Identify the dimension of the region A and find the area

Length of rectangle = $x+8$

Width of the rectangle = $4$

Area of region A= $4\left(x+8\right)$

Identify the dimension of the region B and find the area

Length of rectangle = $x+4+x$

Width of the rectangle = $x$

Area of region B= $x\left(x+4+x\right)$

Area of the shaded region is found by adding the area of region A and area of region B

  $4\left(x+8\right)+x\left(x+4+x\right)$

Distribute the 4 and x in the expression $4\left(x+8\right)+x\left(x+4+x\right)$

  $4x+32+x^2+4x+x^2$

Group like terms

  $x^2+x^2+4x+4x+32$

Combine like terms

  $2x^2+8x+32$

Conclusion:

The area of the shaded region given is $\boxed{2x^2+8x+32}$