class number, divisibility, diophantine equation.

Question

How to generalize or prove the following on class numbers of quadratic fields.

1. Let $n$ be a positive integer then $h(-4n) = 1$ if and only if $n = 1, 2, 3, 4$ or $7$.
2. If $D$ is Gauss discriminant then for $D = b^2 - 4ac < -7$ is true.
3. The Diophantine equation $2x(x^3 + 1) = y^2$ has only solutions in integers are $(0, 0)$, $(-1,0)$, $(1, 2)$, $(1, -2)$, $(2, 6)$ and $(2, -6)$. How to conclude that, there is no solutions of this equation except these I listed now.

Thanking you.