Let the foot of perpendicular from the point A(4, 3, 1) on the plane P: $$x - y + 2z + 3 = 0$$ be N. If $$B(5, \alpha, \beta)$$, $$\alpha, \beta \in \mathbb{Z}$$ is a point on plane P such that the area of the triangle ABN is $$3\sqrt{2}$$, then $$\alpha^2 + \beta^2 + \alpha\beta$$ is equal to _______.

Given point $$A(4, 3, 1)$$, the plane $$P: x - y + 2z + 3 = 0$$ and a point $$B(5, \alpha, \beta)$$ on $$P$$, the foot of the perpendicular from $$A$$ to $$P$$ is $$N$$ and the area of $$\triangle ABN$$ equals $$3\sqrt{2}$$. We seek $$\alpha^2 + \beta^2 + \alpha\beta\,$$.

The normal vector to the plane is $$\vec{n} = (1, -1, 2)$$ with $$|\vec{n}|^2 = 6$$. The parameter for the foot is $$t = \dfrac{\vec{A}\cdot\vec{n} + d}{|\vec{n}|^2} = \dfrac{4 - 3 + 2(1) + 3}{6} = 1\,, $$ so $$N = A - t\,\vec{n} = (4-1,\;3+1,\;1-2) = (3, 4, -1)\,, $$ which indeed satisfies $$3 - 4 + 2(-1) + 3 = 0\,$$.

The vector $$\vec{AN} = (1, -1, 2)$$ has length $$|AN| = \sqrt{6}$$. Since $$B(5,\alpha,\beta)$$ lies on the plane we impose $$5 - \alpha + 2\beta + 3 = 0 \quad\Longrightarrow\quad \alpha = 2\beta + 8\,, $$ so $$B = (5,\,2\beta+8,\,\beta)\,$$.

The area condition $$\tfrac12\,|\vec{AN}\times\vec{BN}| = 3\sqrt{2}$$ becomes $$|\vec{AN}\times\vec{BN}| = 6\sqrt{2}$$. With $$\vec{BN} = N - B = (-2,\,-2\beta - 4,\,-1 - \beta)$$ one computes the determinant

$$\vec{AN}\times\vec{BN} =\begin{vmatrix}\vec{i}&\vec{j}&\vec{k}\\1&-1&2\\-2&-2\beta-4&-1-\beta\end{vmatrix} =(5\beta+9,\;\beta-3,\;-2\beta-6)\,.$$

Its squared magnitude is $$(5\beta+9)^2 + (\beta-3)^2 + (-2\beta-6)^2 =25\beta^2+90\beta+81 + \beta^2-6\beta+9 + 4\beta^2+24\beta+36 =30\beta^2+108\beta+126,$$ so $$30\beta^2+108\beta+126=72$$ gives $$5\beta^2+18\beta+9=0$$. Hence $$\beta = \dfrac{-18\pm\sqrt{324-180}}{10} = \dfrac{-18\pm12}{10} \;\Longrightarrow\; \beta = -\tfrac{3}{5}\ \text{or}\ \beta = -3\,.$$

Selecting the integer solution: Requiring $$\alpha,\beta\in\mathbb{Z}$$ yields $$\beta = -3$$, then $$\alpha = 2(-3)+8 = 2\,$$.

Finally, $$\alpha^2 + \beta^2 + \alpha\beta = 2^2 + (-3)^2 + 2(-3) = 4 + 9 - 6 = 7\,.$$ Therefore, the answer is 7.