1. **State the problem:** We have a compound L-shaped figure made of two rectangles. The total area is 188 m². We need to find the value of $x$ given the side lengths in terms of $x$. 2. **Identify the rectangles:** - Rectangle 1 (left vertical part): height = $4x + 3$, width = $2x + 2$ - Rectangle 2 (top right part): height = $3x$, width = $x - 2$ 3. **Calculate the area of each rectangle:** - Area of Rectangle 1 = height × width = $(4x + 3)(2x + 2)$ - Area of Rectangle 2 = height × width = $(3x)(x - 2)$ 4. **Write the total area equation:** $$ (4x + 3)(2x + 2) + 3x(x - 2) = 188 $$ 5. **Expand the terms:** $$ (4x)(2x) + (4x)(2) + 3(2x) + 3(2) + 3x^2 - 6x = 188 $$ $$ 8x^2 + 8x + 6x + 6 + 3x^2 - 6x = 188 $$ 6. **Simplify the expression:** $$ 8x^2 + 3x^2 + (8x + 6x - 6x) + 6 = 188 $$ $$ 11x^2 + 8x + 6 = 188 $$ 7. **Bring all terms to one side:** $$ 11x^2 + 8x + 6 - 188 = 0 $$ $$ 11x^2 + 8x - 182 = 0 $$ 8. **Solve the quadratic equation using the quadratic formula:** $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \quad \text{where } a=11, b=8, c=-182 $$ Calculate the discriminant: $$ \Delta = 8^2 - 4 \times 11 \times (-182) = 64 + 8008 = 8072 $$ Calculate the roots: $$ x = \frac{-8 \pm \sqrt{8072}}{22} $$ $$ \sqrt{8072} \approx 89.82 $$ So, $$ x_1 = \frac{-8 + 89.82}{22} = \frac{81.82}{22} \approx 3.719 $$ $$ x_2 = \frac{-8 - 89.82}{22} = \frac{-97.82}{22} \approx -4.446 $$ 9. **Interpret the solution:** Since lengths must be positive, we discard $x_2$. 10. **Final answer:** $$ \boxed{x = 3.72} $$ (to 3 significant figures)