Roots and roots

$\begin{aligned} X & = \left({\color{#3D99F6}\sqrt 5+\sqrt 6 }+{\color{#D61F06}\sqrt 7}\right)\left({\color{#3D99F6}\sqrt 5+\sqrt 6 }-{\color{#D61F06}\sqrt 7}\right)\left({\color{#D61F06}\sqrt 5-\sqrt 6 }+{\color{#3D99F6}\sqrt 7}\right)\left({\color{#D61F06}-\sqrt 5+\sqrt 6 }+{\color{#3D99F6}\sqrt 7}\right) & \small \color{#3D99F6} \text{Note that }(a+{\color{#D61F06}b})(a - {\color{#D61F06}b}) = a^2 - {\color{#D61F06} b^2} \\ & = \left({\color{#3D99F6}(\sqrt 5+\sqrt 6)^2}-{\color{#D61F06}7}\right) \left({\color{#3D99F6}7}-{\color{#D61F06}(\sqrt 5-\sqrt 6)^2}\right) \\ & = \left(11 + 2 \sqrt {30}- 7 \right) \left(7-11 + 2\sqrt{30}\right) \\ & = \left(2 \sqrt {30} + 4 \right) \left(2\sqrt{30}-4\right) \\ & = 120 - 16 \\ & = \boxed{104} \end{aligned}$

$(a+b+c)(a+b-c)(a-b+c)(-a+b+c)=2(a^2b^2+b^2c^2+c^2a^2)-(a^4+b^4+c^4)$ (Very similar with the Heron's formula)

So the expression equals to $2(5\times6+6\times7+7\times5)-(5^2+6^2+7^2)=104$