<s>[INST] If $x^4+12x-5$ has roots $x_1,x_2,x_3,x_4$ find polynomial with roots $x_1+x_2,x_1+x_3,x_1+x_4,x_2+x_3,x_2+x_4,x_3+x_4$ I have the polynomial $x^4+12x-5$ with the roots $x_1,x_2,x_3,x_4$ and I want to find the polynomial whose roots are $x_1+x_2,x_1+x_3,x_1+x_4,x_2+x_3,x_2+x_4,x_3+x_4$.
I found the roots $x_1=-1+\sqrt{2},x_2=-1-\sqrt{2},x_3=1-2i,x_4=1+2i$. And after long computations the polynomial is $x^6+20x^2-144$. Are there clever way to find it?
 [/INST] Eliminate $x$ out of the system
$$s^4 + 12 s - 15 = 0 \\
t^4 + 12 t - 15 =0 \\
x- (s+t) = 0$$
and get an equation of degree $10$ in $x$
$$x^{10} + 96 x^7 - 60 x^6 - 144 x^4 + 1920 x^3 - 1600 x^2 - 13824 x + 11520=0 \ \ (*)$$
Now, some of these solutions  ($4$ of them) of the above are in fact solution of the equation in $x$ obtained from
$$s^4 + 12 s - 15 = 0\\
x - (s+s) = 0$$
that is
$$x^4 + 96 x - 80=0 \ \ (**)$$
Divide the polynomials $(*)$ by $(**)$ and get
$$ x^6 + 20 x^2 - 144=(x-2)(x+2)(x^4 + 4 x^2 + 36) $$
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