1. **State the problem:** We have an outer rectangle with dimensions length $3x - 2$ and width $x + 6$, and an inner rectangle with dimensions length $2x$ and width $x - 1$. The shaded region is the area of the outer rectangle minus the area of the inner rectangle, which equals 103 ft². 2. **Write the formula for the shaded area:** $$\text{Shaded area} = \text{Area of outer rectangle} - \text{Area of inner rectangle}$$ 3. **Express the areas in terms of $x$:** $$\text{Area of outer rectangle} = (3x - 2)(x + 6)$$ $$\text{Area of inner rectangle} = (2x)(x - 1)$$ 4. **Set up the equation:** $$ (3x - 2)(x + 6) - 2x(x - 1) = 103 $$ 5. **Expand the terms:** $$ (3x - 2)(x + 6) = 3x \cdot x + 3x \cdot 6 - 2 \cdot x - 2 \cdot 6 = 3x^2 + 18x - 2x - 12 = 3x^2 + 16x - 12 $$ $$ 2x(x - 1) = 2x^2 - 2x $$ 6. **Substitute back into the equation:** $$ 3x^2 + 16x - 12 - (2x^2 - 2x) = 103 $$ 7. **Simplify the left side:** $$ 3x^2 + 16x - 12 - 2x^2 + 2x = 103 $$ $$ (3x^2 - 2x^2) + (16x + 2x) - 12 = 103 $$ $$ x^2 + 18x - 12 = 103 $$ 8. **Bring all terms to one side:** $$ x^2 + 18x - 12 - 103 = 0 $$ $$ x^2 + 18x - 115 = 0 $$ 9. **Solve the quadratic equation using the quadratic formula:** $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ where $a=1$, $b=18$, and $c=-115$. 10. **Calculate the discriminant:** $$ \Delta = 18^2 - 4 \cdot 1 \cdot (-115) = 324 + 460 = 784 $$ 11. **Find the roots:** $$ x = \frac{-18 \pm \sqrt{784}}{2} = \frac{-18 \pm 28}{2} $$ 12. **Calculate each root:** - $$ x = \frac{-18 + 28}{2} = \frac{10}{2} = 5 $$ - $$ x = \frac{-18 - 28}{2} = \frac{-46}{2} = -23 $$ 13. **Choose the valid root:** Since dimensions must be positive, $x = 5$. 14. **Find the dimensions of the inner rectangle:** - Length: $$ 2x = 2 \times 5 = 10 $$ - Width: $$ x - 1 = 5 - 1 = 4 $$ **Final answer:** The inside rectangle has dimensions length 10 ft and width 4 ft.