Question

1. write and simplify an expression to represent the area of the figure.

a)
b) write and simplify an expression to represent the area of the shaded region

Explanation:

Step1: Find the area of the large rectangle

The formula for the area of a rectangle is \(A = length\times width\). For the large rectangle, length \(l=(x + 6)\) and width \(w=(x - 3)\). Using the FOIL method \((a + b)(c + d)=ac+ad+bc+bd\), we have \(A_{1}=(x + 6)(x - 3)=x^{2}-3x+6x - 18=x^{2}+3x - 18\).

Step2: Find the area of the small rectangle

For the small rectangle, length \(l = 2x\) and width \(w=(x + 1)\). Using the distributive property \(a(b + c)=ab+ac\), we get \(A_{2}=2x(x + 1)=2x^{2}+2x\).

Step3: Find the area of the shaded region

The area of the shaded region \(A=A_{1}-A_{2}\). Substitute \(A_{1}\) and \(A_{2}\) into the formula:

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Answer:

The simplified expression for the area of the shaded region is \(-x^{2}+x - 18\)