If $$2x^2 + y^2 + 8z^2 - 2\sqrt{2}xy + 4\sqrt{2}yz - 8zx = (Ax + y + Bz)^2,$$ then the value of $$(A^2 + B^2 - AB)$$ is:

$$2x^2 + y^2 + 8z^2 - 2\sqrt{2}xy + 4\sqrt{2}yz - 8zx = (Ax + y + Bz)^2$$

= $$2x^2 + y^2 + 8z^2 + 2(-\sqrt{2}xy - 2\sqrt{2}yz + 4zx) = (Ax + y + Bz)^2,$$

= $$2x^2 + y^2 + 8z^2 + 2(-\sqrt{2}x.y - y.2\sqrt{2}z + \sqrt{2}x.2\sqrt{2}z) = (Ax + y + Bz)^2$$

= $$(+\sqrt{2}x - y + 2\sqrt{2}z)^2 = (Ax + y + Bz)^2$$

Ax = $$\sqrt{2}x$$

A = $$\sqrt{2}$$

y = -1

B = $$2\sqrt{2}$$

Now,

$$(A^2 + B^2 - AB)$$

$$((\sqrt{2})^2 + (2\sqrt{2})^2 - \sqrt{2}.2\sqrt{2})$$

= 2 + 8 - 4 = 6