<s>[INST] is there away to give $= \sqrt[3]{5 + 10i } + \sqrt[3]{5- 10i}$ question is there away to give 
$$= \sqrt[3]{5 + 10i } 
 + \sqrt[3]{5- 10i}$$
in just reals?

I am thinkting maybe use some clever algebra raise it to like 3 and magically imaginary terms dissapear. Or expressed the complex numbers in polar form but idk

Background (might delete if this is not really the question)
Let $$ m(x)=x^3-15x-10 $$
Find the roots of $m(x)$ using the cubic formula and show that they are all real 
Using Howell complex approach 
Lets change variables $$ \begin{aligned}
&z^3-15z-10
\\=&z^3+0z^2-15z-10 
\end{aligned}$$
so, general form is denoted as 
$$ z^3+a_2z^2+a_1z+a_0=0$$
so $$\begin{aligned}
a_2&=0
\\a_1&=-15
\\a_0&=-10
\end{aligned}$$
we get by substituting that $z=x-a_2/3$. $a_2=0$ its already depressed cubic but so theses steps are unnecesary for this one problem but will use them for other problems
$$ x^3+bx+c$$
where $$\begin{aligned}
b&=a_1-\frac{a_2^2}{3}
                       =-15-\frac{0^2}{3}=-15
\\c&=\frac{-a_1a_2}{3}+\frac{2}{27}a_2^3+a_0
                        =\frac{-(-15)(0)}{3}+\frac{2}{27}(0)^3+-10=-10
\end{aligned} $$
our depressed cubic is of $$  x^3-15x-10 $$
Ferro-Tartaglia Formula 
$$ x=\sqrt[3]{\frac{-c}{2} + \sqrt{\frac{c^2}{4} +\frac{b^3}{27}}}\ 
+ \sqrt[3]{\frac{-c}{2} - \sqrt{\frac{c^2}{4} -\frac{b^3}{27}}}\ $$
in our case $$ 
\begin{aligned}
x&=\sqrt[3]{\frac{10}{2} + \sqrt{\frac{100}{4} +\frac{(-15)^3}{27}}}\ 
+ \sqrt[3]{\frac{10}{2} - \sqrt{\frac{100}{4} +\frac{(-15)^3}{27}}}\ 
 \\&= \sqrt[3]{5 + \sqrt{25 -125}} 
 + \sqrt[3]{5- \sqrt{25 -125}}
\\ &=  \sqrt[3]{5 + 10i } 
 + \sqrt[3]{5- 10i}
\end{aligned}
$$
 [/INST] Since
$$
(5+10i)^{1/3}=\sqrt5\,e^{i\arctan(2)/3}
$$
and
$$
(5-10i)^{1/3}=\sqrt5\,e^{-i\arctan(2)/3}
$$
Thus,
$$
\bbox[5px,border:2px solid #C0A000]{(5+10i)^{1/3}+(5-10i)^{1/3}=2\sqrt5\cos\left(\frac{\arctan(2)}3\right)}
$$

Another Approach to the Original Question
Let $\alpha=2\sqrt5$. Then, using $\cos(3x)=4\cos^3(x)-3\cos(x)$, we get
$$
\begin{align}
10
&=\frac{\alpha^3}4\cos(3x)\\
&=\alpha^3\cos^3(x)-\frac34\alpha^3\cos(x)\\[5pt]
&=\underbrace{\alpha^3\cos^3(x)}_{z^3}-15\,\underbrace{\ \alpha\cos(x)\ }_z
\end{align}
$$
Thus, with $\cos(3x)=\frac{40}{\alpha^3}=\frac1{\sqrt5}$, we get that $z=2\sqrt5\cos(x)$ is a root. This gives $3$ real roots
$$
z_1=2\sqrt5\cos\left(\frac13\arccos\left(\frac1{\sqrt5}\right)\right)
$$
$$
z_2=2\sqrt5\cos\left(\frac{2\pi}3+\frac13\arccos\left(\frac1{\sqrt5}\right)\right)
$$
$$
z_3=2\sqrt5\cos\left(\frac{4\pi}3+\frac13\arccos\left(\frac1{\sqrt5}\right)\right)
$$
Noting that $\arccos\left(\frac1{\sqrt5}\right)=\arctan(2)$, we get that $z_1=(5+10i)^{1/3}+(5-10i)^{1/3}$.
</s>