Convex Quadratic?

Geometry Level 3

Consider a curve y=ax2+bx+cy=ax^2 +bx+c with a,b,cN a, b, c \in \mathbb{N} which passes through four points A(2,3),B(1,1),C(α,β),D(2,7)A(-2,3),B(-1,1),C(\alpha ,\beta ),D(2,7).

All these points are taken in given order for constructing a convex quadrilateral, which has maximum possible area.

Find minimum possible value of a+b+c+2α+4β.a+b+c+2\alpha +4\beta.

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