I don't understand how we choose the x-values to solve the equation system proving linear independence. For example I have this question:
Proving the linear dependence is trivial, but then this follows:
So I underand the first part, but how do we choose the three x values, namely x = 0, x = -1 and x = 1?
You can choose arbitrary values for $x$. It's purely for simplification purposes. In this case choosing $x=0$ in $\alpha_1+\alpha_2\cdot 0+\alpha_2 \cdot (1+0+0^2) = 0$ simplifies to $\alpha_1+\alpha_2=0$. The values are usually chosen as a solution to one of the polynomials. In this case $x$ and $x^2+x+1$. The second polynomial doesn't have real solutions so you can't pick a nice number there instead they use $x=1$ and $x=-1$.