<u>Given</u>:
The length of the side of the metal is x + 7.
The length of the side of the hole is x - 2.
We need to determine the area of the metal part or the shaded region.
<u>Area of the metal:</u>
The area of the metal can be determined using the formula,
![A=s^2](https://tex.z-dn.net/?f=A%3Ds%5E2)
Substituting s = x + 7, we get;
![A=(x+7)^2](https://tex.z-dn.net/?f=A%3D%28x%2B7%29%5E2)
![A=x^2+14x+49](https://tex.z-dn.net/?f=A%3Dx%5E2%2B14x%2B49)
Thus, the area of the metal is
square units.
<u>Area of the hole:</u>
The area of the hole can be determined using the formula,
![A=s^2](https://tex.z-dn.net/?f=A%3Ds%5E2)
Substituting s = x -2 ,we get;
![A=(x-2)^2](https://tex.z-dn.net/?f=A%3D%28x-2%29%5E2)
![A=x^2-4x+4](https://tex.z-dn.net/?f=A%3Dx%5E2-4x%2B4)
Thus, the area of the hole is
square units.
<u>Area of the shaded region:</u>
The area of the shaded region can be determined by subtracting the area of the hole from the area of the metal.
Thus, we have;
Area = Area of the metal - Area of the hole
Substituting the values, we have;
![Area = x^2+14x+49-(x^2-4x+4)](https://tex.z-dn.net/?f=Area%20%3D%20x%5E2%2B14x%2B49-%28x%5E2-4x%2B4%29)
Simplifying, we have;
![Area = x^2+14x+49-x^2+4x-4](https://tex.z-dn.net/?f=Area%20%3D%20x%5E2%2B14x%2B49-x%5E2%2B4x-4)
![Area = 18x+45](https://tex.z-dn.net/?f=Area%20%3D%2018x%2B45)
Thus, the area of the shaded region is (18x + 45) square units.