Given:
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.
Area of the metal:
The area of the metal can be determined using the formula,
[tex]A=s^2[/tex]
Substituting s = x + 7, we get;
[tex]A=(x+7)^2[/tex]
[tex]A=x^2+14x+49[/tex]
Thus, the area of the metal is [tex]x^2+14x+49[/tex] square units.
Area of the hole:
The area of the hole can be determined using the formula,
[tex]A=s^2[/tex]
Substituting s = x -2 ,we get;
[tex]A=(x-2)^2[/tex]
[tex]A=x^2-4x+4[/tex]
Thus, the area of the hole is [tex]x^2-4x+4[/tex] square units.
Area of the shaded region:
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;
[tex]Area = x^2+14x+49-(x^2-4x+4)[/tex]
Simplifying, we have;
[tex]Area = x^2+14x+49-x^2+4x-4[/tex]
[tex]Area = 18x+45[/tex]
Thus, the area of the shaded region is (18x + 45) square units.
