Proving some facts about the EC $y^2 = x^3 + ax + b$

Question

A solution to this question would be much appreciated!

If $E/F$ is the EC defined by $y^2 = x^3 + ax + b$ then prove the following:

1. If $P = (x, y)$ is element of $E(F)$ with order 3 then $x$ is a root of the poly $q(x) = 3x^4 + 6ax^2 + 12bx − a^2$

2. Conversely, if $x$ is root of $q(x)$ and if $x^3 + ax + b = y^2$ is square in $F$ then $P = (x, y)$ is a point of order $3$ in $E(F)$

Answer 1

If a point $P=(x,y)$ has order three, then $2P=-P$, so that the duplication formula for the $x$-coordinate gives, with $y^2=f(x)=x^3+ax+b$, and $16(4a^3+27b^2)\neq 0$, $$ 0=\Psi_3(x)=2f(x)f''(x)-f'(x)^2=3x^4 + 4ax^3 + 6ax^2 + 12x -a^2. $$