1. **State the problem:** We have a right triangle with legs $CA = \sqrt{5} - 2$ and $CB = \sqrt{5} + 2$, and we need to find the hypotenuse $AB$. 2. **Formula used:** In a right triangle, the Pythagorean theorem states: $$AB^2 = CA^2 + CB^2$$ 3. **Calculate each leg squared:** $$CA^2 = (\sqrt{5} - 2)^2 = (\sqrt{5})^2 - 2 \times 2 \times \sqrt{5} + 2^2 = 5 - 4\sqrt{5} + 4 = 9 - 4\sqrt{5}$$ $$CB^2 = (\sqrt{5} + 2)^2 = (\sqrt{5})^2 + 2 \times 2 \times \sqrt{5} + 2^2 = 5 + 4\sqrt{5} + 4 = 9 + 4\sqrt{5}$$ 4. **Sum the squares:** $$AB^2 = (9 - 4\sqrt{5}) + (9 + 4\sqrt{5}) = 9 + 9 + (-4\sqrt{5} + 4\sqrt{5}) = 18 + 0 = 18$$ 5. **Find the hypotenuse:** $$AB = \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$$ **Final answer:** The length of the hypotenuse $AB$ is $3\sqrt{2}$.