The mean proportion between (3 + $\sqrt{2}$) and (12 − $\sqrt{32}$) is:

1. 6

2. $2\sqrt{7}$

3. $\sqrt{7}$

4. $\dfrac{15 - 3\sqrt{2}}{2}$

Let mean proportion be x.

(3 + $\sqrt{2}$) : x = x : (12 − $\sqrt{32}$)

$\Rightarrow \dfrac{(3 + \sqrt{2})}{x} = \dfrac{x}{(12 − \sqrt{32})} \\[1em] \Rightarrow x^2 = (3 + \sqrt{2}) \times (12 − \sqrt{32}) \\[1em] \Rightarrow x^2 = (3 + \sqrt{2}) \times 4(3 − \sqrt{2}) \\[1em] \Rightarrow x^2 = 4[(3)^2 − (\sqrt{2})]^2 \\[1em] \Rightarrow x^2 = 4(9 - 2) \\[1em] \Rightarrow x^2 = 4 \times 7 \\[1em] \Rightarrow x^2 = 28 \\[1em] \Rightarrow x = \sqrt{28} \\[1em] \Rightarrow x = 2\sqrt7.$

Hence, option 2 is the correct option.