Answer
$\left\{-4,-\dfrac{1}{2}\right\}$
Work Step by Step
Using factoring of trinomials, the given equation, $
8x^6+513x^3+64=0
,$ is equivalent to
\begin{align*}
(x^3+64)(8x^3+1)&=0
.\end{align*}
Equating each factor to zero (Zero Product Property) and solving for the variable, then
\begin{array}{l|r}
x^3+64=0 & 8x^3+1=0
\\
x^3=-64 & 8x^3=-1
\\
& x^3=-\dfrac{1}{8}
.\end{array}
Taking the cube root of both sides, the equations above are equivalent to
\begin{array}{l|r}
x=\sqrt[3]{-64} & m=\sqrt[3]{-\dfrac{1}{8}}
\\\\
x=\sqrt[3]{(-4)^3} & m=\sqrt[3]{\left(-\dfrac{1}{2}\right)^3}
\\\\
x=-4 & m=-\dfrac{1}{2}
.\end{array}
Hence, the real solutions of the equation $
2m^6+11m^3+5=0
$ is the set $\left\{-4,-\dfrac{1}{2}\right\}$.
