Questions: Find the Area of the shaded rectangular region.

Solution Steps

The dimensions of the larger rectangle are given as \(5x - 2\) for the length and \(2x + 3\) for the width. The area \(A_{\text{large}}\) of a rectangle is calculated by multiplying the length and the width: \[ A_{\text{large}} = (5x - 2)(2x + 3) \]

Expand the expression using the distributive property: \[ A_{\text{large}} = (5x - 2)(2x + 3) \] \[ A_{\text{large}} = 5x \cdot 2x + 5x \cdot 3 - 2 \cdot 2x - 2 \cdot 3 \] \[ A_{\text{large}} = 10x^2 + 15x - 4x - 6 \] \[ A_{\text{large}} = 10x^2 + 11x - 6 \]

The dimensions of the smaller rectangle are given as \(x\) for the length and \(3x + 1\) for the width. The area \(A_{\text{small}}\) of the smaller rectangle is: \[ A_{\text{small}} = x(3x + 1) \] \[ A_{\text{small}} = 3x^2 + x \]

The area of the shaded region is the difference between the area of the larger rectangle and the area of the smaller rectangle: \[ A_{\text{shaded}} = A_{\text{large}} - A_{\text{small}} \] \[ A_{\text{shaded}} = (10x^2 + 11x - 6) - (3x^2 + x) \] \[ A_{\text{shaded}} = 10x^2 + 11x - 6 - 3x^2 - x \] \[ A_{\text{shaded}} = 7x^2 + 10x - 6 \]

Final Answer