<s>[INST] How find the $(x^2_{1}+x_{1}x_{2}+x^2_{2})(x^2_{2}+x_{2}x_{3}+x^3_{3})(x^2_{3}+x_{3}x_{1}+x^2_{1})$ Let  $x_1$, $x_2$, and $x_3$ be the roots of the equation
$$4x^3-6x^2+7x-8=0.$$
Find this value:
$$(x_1^2+x_1x_2+x_2^2)(x_2^2+x_2x_3+x_3^3)(x_3^2+x_3x_1+x_1^2)$$
My try:
$$x_1+x_2+x_3=\frac 3 2$$
$$x_1 x_2+x_2 x_3+x_1 x_3=\frac 7 4$$
$$x_1 x_2 x_3=2$$
Now I have
$$x_1^2+x_1x_2+x_2^2=(x_1+x_2)^2-x_1x_2=(\frac 3 2-x_3)^2-\frac 2 {x_3}.$$
But this is very ugly, and I think this problem should have a cleaner solution. Thanks.
 [/INST] Notice
$$
x_1^2 + x_1x_2 + x_2^2 = \frac{x_1^3 - x_2^3}{x_1 - x_2}
= \frac{(6x_1^2 -7x_1 + 8)-(6x_2^2 -7x_2 + 8)}{4(x_1-x_2)}\\
= \frac{6(x_1+x_2)-7}{4} = \frac{6(\frac{3}{2} - x_3) - 7}{4}
= \frac{1-3x_3}{2} = \frac32(\frac13 - x_3)
$$
and similar expression holds for $x_2^2 + x_2x_3 + x_3^2$ and $x_3^2 + x_3x_1 + x_1^2$.
The expression we want is equal to
$$\begin{align}
  & \left(\frac{3}{2}\right)^3(\frac13 - x_1)(\frac13 - x_2)(\frac13 - x_3)\\
= & \frac14 \left(\frac{3}{2}\right)^3\left[4\left(\frac13\right)^3 - 6\left(\frac13\right)^2 + 7\left(\frac13\right) - 8\right]\\
= & -\frac{167}{32}
\end{align}$$
</s>