1. **State the problem:** We need to find the base and height of a triangular window where the area is 120 square feet, and the base is 4 feet more than twice the height. 2. **Define variables:** Let the height be $h$ feet. 3. **Express the base in terms of height:** The base $b$ is given by $b = 2h + 4$. 4. **Write the area formula for a triangle:** The area $A$ is given by $$A = \frac{1}{2} \times b \times h$$ 5. **Substitute the known values:** We know $A = 120$ and $b = 2h + 4$, so $$120 = \frac{1}{2} \times (2h + 4) \times h$$ 6. **Simplify the equation:** $$120 = \frac{1}{2} (2h^2 + 4h) = h^2 + 2h$$ 7. **Rewrite as a quadratic equation:** $$h^2 + 2h - 120 = 0$$ 8. **Solve the quadratic equation:** Using the quadratic formula $h = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ with $a=1$, $b=2$, $c=-120$, $$h = \frac{-2 \pm \sqrt{2^2 - 4 \times 1 \times (-120)}}{2} = \frac{-2 \pm \sqrt{4 + 480}}{2} = \frac{-2 \pm \sqrt{484}}{2} = \frac{-2 \pm 22}{2}$$ 9. **Calculate the two possible values:** - $h = \frac{-2 + 22}{2} = \frac{20}{2} = 10$ - $h = \frac{-2 - 22}{2} = \frac{-24}{2} = -12$ (not physically possible since height cannot be negative) 10. **Find the base:** For $h=10$, $$b = 2(10) + 4 = 20 + 4 = 24$$ **Final answer:** The height is 10 feet and the base is 24 feet.