I have the following polynomial optimization problem in $x \in \mathbb{R}$.
$$\begin{array}{ll} \text{minimize} & x^3+9x^2+24x+15\\ \text{subject to} & x+3 \geq 0\end{array} \tag{$P$}$$
It is not hard to see that the solution is $x^*=-2$.
I now have some exercises, where I am supposed to transform the polynomial optimization problem above into a linear one. First, I have to add redundant inequalities to it, which in this case are
$$(ax+b)^2 \geq 0, \qquad (ax+b)^2(x+3) \geq 0$$
for all $a, b \in \mathbb{R}$. Then I replace any appearance of $x^2$ and $x^3$ by $y$ and $z$, so we get the following (infinite) linear optimization problem in $x, y, z \in \mathbb{R}$.
$$\begin{array}{ll} \text{minimize} & z+9y+24x+15\\ \text{subject to} & x+3 \geq 0\\ & a^2y+2abx+b^2 \geq 0\\ & a^2z+(2ab+3a^2)y+(b^2+6ab)x+3b^2 \geq 0, \qquad \forall a, b \in \mathbb{R}\end{array} \tag{$P_1$}$$
However, now our problem has infinitely many conditions, so what we need to do is find out which conditions are active in the solution $(x^*,y^*,z^*) = (-2,4,-8)$ and then single out a finite amount of those, without weakening the original problem. My exercises are roughly stated as follows:
Find all the conditions in ($P_1$) that are active in $(x^*,y^*,z^*) = (-2,4,-8)$.
Use $1.$ to find a finite amount of conditions to create a (finite) linear optimization problem ($P_2$) so that the solution of $P_2$ is still $x^*=-2$.
I am not sure how to tackle either problem, so I would be grateful if someone could help out.