We want to find real roots for the following function
[tex]f(x)=6(x^2+6)(x^2+8)^2[/tex]
The roots of this function are the solutions for the following equation
[tex]6(x^2+6)(x^2+8)^2=0[/tex]
Since we have a product of two terms equals zero, this means that at least one of them is zero.
[tex]\begin{gathered} 6(x^2+6)(x^2+8)^2=0 \\ (x^2+6)(x^2+8)^2=0\Rightarrow\begin{cases}(x^2+6)=0 \\ (x^2+8)^2=0\end{cases} \end{gathered}[/tex]
Solving those two equations:
[tex]\begin{cases}(x^2+6)=0 \\ (x^2+8)^2=0\end{cases}\Rightarrow\begin{cases}x^2=-6 \\ x^2=-8\end{cases}\Rightarrow\begin{cases}x^{}=\pm\sqrt[]{6}i \\ x^{}=\pm2\sqrt[]{2}i\end{cases}[/tex]
This function doesn't have real solutions.
