Question

in the accompanying diagram the width of the inner rectangle is represented by $x - 3$, and the length by $x + 3$. the width of the outer rectangle is represented by $3x - 4$, and the length by $3x + 4$.

express the area of the shaded region as a polynomial in expanded form in terms of $x$.

Explanation:

Step1: Calculate the area of the outer rectangle

The area formula for a rectangle is \(A = length\times width\). For the outer rectangle, length \(l = 3x + 4\) and width \(w=3x - 4\). Using the formula \((a + b)(a - b)=a^{2}-b^{2}\), we have \(A_{outer}=(3x + 4)(3x - 4)=9x^{2}-16\).

Step2: Calculate the area of the inner rectangle

For the inner rectangle, length \(l = x + 3\) and width \(w=x - 3\). Using the formula \((a + b)(a - b)=a^{2}-b^{2}\), we get \(A_{inner}=(x + 3)(x - 3)=x^{2}-9\).

Step3: Calculate the area of the shaded region

The area of the shaded region \(A = A_{outer}-A_{inner}\). Substitute the expressions for \(A_{outer}\) and \(A_{inner}\): \(A=(9x^{2}-16)-(x^{2}-9)\). Expand the expression: \(A = 9x^{2}-16 - x^{2}+9\). Combine like - terms: \(A=(9x^{2}-x^{2})+(9 - 16)=8x^{2}-25\).

Answer:

\(8x^2 - 25\)