Answer: 104

Small Hint:

Pair factors so that each pair is a difference of squares

Big Hint:

After the first pairing, the remaining radicals occur only through $\sqrt{42}$

Solution:

Pair the first two factors, then the last two: $$\begin{gathered} (\sqrt6+\sqrt7)^2-(\sqrt5)^2\\ {}=8+2\sqrt{42},\\ (\sqrt5)^2-(\sqrt7-\sqrt6)^2\\ {}=-8+2\sqrt{42}. \end{gathered}$$ Therefore the requested product is $$\begin{aligned} &(8+2\sqrt{42})(-8+2\sqrt{42})\\ &\qquad=168-64=104. \end{aligned}$$