We are asked to evaluate the expression $(3\sqrt{7} - \sqrt{8})^2$.

1. Problem Description

We are asked to evaluate the expression $(3\sqrt{7} - \sqrt{8})^2$.

2. Solution Steps

We can use the formula $(a-b)^2 = a^2 - 2ab + b^2$ to expand the expression.

$(3\sqrt{7} - \sqrt{8})^2 = (3\sqrt{7})^2 - 2(3\sqrt{7})(\sqrt{8}) + (\sqrt{8})^2$

$= 9 \cdot 7 - 6\sqrt{7 \cdot 8} + 8$

$= 63 - 6\sqrt{56} + 8$

$= 71 - 6\sqrt{4 \cdot 14}$

$= 71 - 6(2\sqrt{14})$

$= 71 - 12\sqrt{14}$

3. Final Answer

$71 - 12\sqrt{14}$