<s>[INST] $x=\sqrt[3]{\sqrt{5}+2}+\sqrt[3]{\sqrt{5}-2}$ is rational or irrational? 
The number $x$ defined below is rational or irrational?
  $$x=\sqrt[3]{\sqrt{5}+2}+\sqrt[3]{\sqrt{5}-2}$$
  From: IMO 1973 - Longlist

My attempt (my real question is at the end): 
the identity $a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^3-ab-ac-bc)$ when $a+b+c=0$, leads to $$a^3+b^3+c^3=3abc \tag{1}$$ 
Now considering
$$a=\sqrt[3]{\sqrt{5}+2},b=\sqrt[3]{\sqrt{5}-2},c=-x$$
from (1) it is true that 
$$x^3-3x-2\sqrt{5}=0 \tag{2}$$
That is the number $x$ is a root from (2). 
Note: By trial and error I've found that answer is $x=\sqrt{5}$ (the other 2 roots are complex), that is irrational. But my question is more subtle.
Question: Can I conclude just inspecting (2), judging by the coefficient $2\sqrt{5}$, that $x$ is irrational, without actually solving the equation? In a math contest that might be helpful, if possible, as it would avoid extra steps. 
 [/INST] Alternatively
$$x=\sqrt[3]{\sqrt{5}+2}+\sqrt[3]{\sqrt{5}-2}=\\
\sqrt[3]{\frac{8\sqrt{5}+16}{8}}+\sqrt[3]{\frac{8\sqrt{5}-16}{8}}=\\
\sqrt[3]{\frac{5\sqrt{5}+3\cdot5+3\sqrt{5}+1}{8}}+\sqrt[3]{\frac{5\sqrt{5}-3\cdot5+3\sqrt{5}-1}{8}}=\\
\sqrt[3]{\frac{\left(\sqrt{5}+1\right)^3}{2^3}}+\sqrt[3]{\frac{\left(\sqrt{5}-1\right)^3}{2^3}}=\\
\frac{\sqrt{5}+1}{2}+\frac{\sqrt{5}-1}{2}=\\
\sqrt{5}$$ 
is irrational.
</s>