Explanation:

- To find the positive square root of \(69 + 28\sqrt{5}\), we assume it can be expressed as \((a + b\sqrt{5})^2\).

- Expanding this: \((a + b\sqrt{5})^2 = a^2 + 2ab\sqrt{5} + 5b^2\).

- Equate it to \(69 + 28\sqrt{5}\).

- \(a^2 + 5b^2 = 69\)

- \(2ab = 28\)

- Solving \(2ab = 28\) gives \(ab = 14\).

- Solving equations, \(a = 7\) and \(b = 2\) works, so \((7 + 2\sqrt{5})^2 = 69 + 28\sqrt{5}\).

Correct Answer: Option 1, \(7 + 2\sqrt{5}\)