(c)

Now that we know a + b = p/n 2sqrt(3), and that a - b = 4, we can use system of equations to get the individual values of each variable.

__Case 1:__ a + b = 2sqrt(3)

a + b = 2sqrt(3)

a - b = 4

Adding both equations, 2a = 4 + 2sqrt(3), a = 2 + sqrt(3).

Plugging in 2 + sqrt(3) for the top equation, 2 + sqrt(3) - 2sqrt(3) = -b, so b = sqrt(3) - 2.

__So for case 1__, \(a = 2 + \sqrt{3}\), and \(b = -2 + \sqrt{3}\)

__Case 2:__ a + b = -2sqrt(3)

a + b = -2sqrt(3)

a - b = 4

Adding both equations, 2a = 4 - 2sqrt(3), a = 2 - sqrt(3)

Plugging in 2 - sqrt(3) for the top equation, 2 - sqrt(3) + 2sqrt(3) = -b, so b = -2 - sqrt(3).

__So for case 2__, \(a = 2 - \sqrt{3}\), and \(b = -2 - \sqrt{3}\)

Hence, we have:

a = 2 + sqrt(3) => b = -2 + sqrt(3)

or

a = 2 - sqrt(3) => b = -2 - sqrt(3)