1. The problem asks to show that the total area of the shaded regions inside the rectangle is $18x - 30$ cm². 2. First, find the area of the rectangle. Its width is $x + 6$ and height is $3x - 5$, so the area $A_{rectangle}$ is: $$A_{rectangle} = (x + 6)(3x - 5)$$ 3. Expand the product: $$A_{rectangle} = x(3x - 5) + 6(3x - 5) = 3x^2 - 5x + 18x - 30 = 3x^2 + 13x - 30$$ 4. Next, find the area of the triangle inside the rectangle. The base of the triangle is $2x$ and its height is equal to the height of the rectangle, $3x - 5$. 5. The area of a triangle is given by: $$A_{triangle} = \frac{1}{2} \times base \times height = \frac{1}{2} \times 2x \times (3x - 5)$$ 6. Simplify the triangle area: $$A_{triangle} = x(3x - 5) = 3x^2 - 5x$$ 7. The shaded regions consist of the part of the rectangle not occupied by the triangle, so their total area is: $$A_{shaded} = A_{rectangle} - A_{triangle} = (3x^2 + 13x - 30) - (3x^2 - 5x)$$ 8. Simplify by subtracting: $$A_{shaded} = 3x^2 + 13x - 30 - 3x^2 + 5x = (3x^2 - 3x^2) + (13x + 5x) - 30 = 18x - 30$$ 9. Thus, the total area of the shaded regions is $18x - 30$ cm², as required.