Question 1: Express F(x) as the sum of a squared quadratic minus a line. The line will be the tangent line and the roots of the quadratic are the x coordinate locations of the two tangent points.
F(x) = ([sqrt(a)]*x^2 + [b/(2*sqrt(a))]*x + [(4ac-b^2)/(8*sqrt(a^3))])^2 - ([-d+b*(4ac-b^2)/(8a^2))]*x + [-e+b^2*(4ac-b^2)^2/(64a^3)])

For F(x) to have a double tangent the quadratic must have two real roots, that means its discriminant must be greater than 0:
[b/(2*sqrt(a))]^2 - 4*[sqrt(a)]*[(4ac-b^2)/(8*sqrt(a^3))] > 0
This simplifies to (3b^2 - 8ac)/(4a) > 0. Since A is nonzero then 3b^2 - 8ac > 0.

Question 2: The second derivative of F(x) is F"(x) = 12ax^2 + 6bx + 2c. To have two distinct inflection points the second derivative must have two distinct real roots. Since F"(x) is quadratic this implies its discriminant must be greater than 0:
(6b)^2 - 4*(12a)*(2c) > 0.
This simplifies to 3b^2 - 8ac > 0.

In both cases the necessary condition is 3b^2 - 8ac > 0. This implies the two conditions are equivalent.

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