Question

write a polynomial that describes the area of the shaded region
the area of the shaded region is
(simplify your answer. type an expression using x as the variable.)

Explanation:

Step1: Calculate the area of the large square

The area of a square is \(A = s^2\), where \(s\) is the side - length. For the large square with side - length \(s=x + 6\), the area \(A_{1}=(x + 6)^2\). Using the formula \((a + b)^2=a^{2}+2ab + b^{2}\) (where \(a=x\) and \(b = 6\)), we have \(A_{1}=x^{2}+12x + 36\).

Step2: Calculate the area of the small square

The area of a square is \(A=s^2\). For the small square with side - length \(s = 2\), the area \(A_{2}=2^{2}=4\).

Step3: Calculate the area of the shaded region

The area of the shaded region \(A\) is the area of the large square minus the area of the small square. So \(A=A_{1}-A_{2}\). Substitute \(A_{1}=x^{2}+12x + 36\) and \(A_{2}=4\) into the formula: \(A=(x^{2}+12x + 36)-4\).
Simplify the expression: \(A=x^{2}+12x+(36 - 4)=x^{2}+12x + 32\).

Answer:

\(x^{2}+12x + 32\)