Let $a$ be the real root of the equation $x^3+x+1=0$
Calculate $$\sqrt[\leftroot{-2}\uproot{2}3]{{(3a^{2}-2a+2)(3a^{2}+2a)}}+a^{2}$$
The correct answer should be $ 1 $. I've tried to write $a^3$ as $-a-1$ but that didn't too much, I guess there is some trick here :s
On
The expansion of what is under square root gives
$$A=9a^4+6a^3-6a^3-4a^2+6a^2+4a=$$ $$a (9a^3+2a+4)=a (9 (-a-1)+2a+4)=$$ $$-a (7a+5)=$$
this must be equal to $$B=(1-a^2)^3=1-3a^2+3a^4-a^6=$$ $$$$
the difference is $$B-A=1+4a^2+5a+3a^4-(a+1)^2=$$ $$3a^2+3a+3a^4=3a (a^3+a+1)=0$$
It is correct.
On
The expression under the cube root is $$9a^4+2a^2+4a=9a(-(a+1))+2a^2+4a=-7a^2-5a$$
Meanwhile consider $$(1-a^2)^3=1-3a^2+3a^4-a^6=1-3a^2+3a(-a-1)-a^2-2a-1$$ $$=-7a^2-5a$$
So the required result follows immediately
On
The real root $a$ of $x^3+x+1$ belongs to the interval $\left(-1,-\frac{1}{2}\right)$ and in order to check that $$ \sqrt[3]{(3a^2-2a+2)(3a^2+2a)}=1-a^2 \tag{1}$$ holds it is enough to check that $$ (3a^2-2a+2)(3a^2+2a)=1-3a^2+3a^4+a^6 \tag{2}$$ holds, or that $$ (1+a+a^3)(-1+5a+a^3) = 0 \tag{3} $$ holds, but that is trivial.
Maybe there is a trick to it, but good ol' factoring can work here.
It is easy to check (using the identity $a^3=-a-1$) that $$ (3a^2-2a+2)(3a^2+2a) = -5a-7a^2 $$ Let $\sqrt[3]{-5a-7a^2}+a^2 = y$. Given that $a\in \mathbb{R}$, we know that also $y\in \mathbb{R}$, and we must solve for it. $$y-a^2 = \sqrt[3]{-5a-7a^2} \\ (y-a^2)^3 = -5a-7a^2 \\ y^3 - 3y^2a^2 + 3ya^4-a^6 = -5a-7a^2 \\ y^3-3y^2a^2+3ya(-a-1)-(-a-1)^2=-5a-7a^2 \\ y^3-3y^2a^2-3ya^2-3ya+6a^2+3a-1=0$$ So many threes, only the $6$ stands out. But if we use $6=3+3$: $$ (y^3-1)+(3a^2-3y^2a^2)+(3a^2+3a-3ya^2-3ya)=0 \\ (y^3-1)-3a^2(y^2-1)-(3a^2+3a)(y-1)=0$$ As one can see, it is possible to factor out $(y-1)$, thereby concluding that $y=1$ is a solution.