Question

write a polynomial in standard form that represents the area of the shaded region. diagram: a square with side length ( x + 1 ), containing an inner rectangle with length ( x - 7 ) and width 5 the area is (square).

Explanation:

Step1: Find area of outer square

The outer figure is a square with side length \( x + 1 \). The area of a square is side length squared, so the area of the outer square is \( (x + 1)^2 \). Using the formula \( (a + b)^2 = a^2 + 2ab + b^2 \), we expand this to \( x^2 + 2x + 1 \).

Step2: Find area of inner rectangle

The inner figure is a rectangle with length \( x - 7 \) and width \( 5 \). The area of a rectangle is length times width, so the area of the inner rectangle is \( 5(x - 7) \). Distributing the 5, we get \( 5x - 35 \).

Step3: Subtract inner area from outer area

To find the area of the shaded region, we subtract the area of the inner rectangle from the area of the outer square. So we have \( (x^2 + 2x + 1) - (5x - 35) \).

Step4: Simplify the expression

First, distribute the negative sign: \( x^2 + 2x + 1 - 5x + 35 \). Then combine like terms: \( x^2 + (2x - 5x) + (1 + 35) = x^2 - 3x + 36 \).

Answer:

\( x^2 - 3x + 36 \)