\(\Leftrightarrow\left(x^2+4x+8\right)^2+3x\left(x^2+4x+8\right)+\frac{9}{4}x^2-\frac{1}{4}x^2=0\)
\(\Leftrightarrow\left(x^2+4x+8+\frac{3}{2}x\right)^2-\frac{1}{4}x^2=0\)
\(\Leftrightarrow\left(x^2+\frac{11}{2}x-\frac{1}{2}x+8\right)\left(x^2+\frac{11}{2}x+\frac{1}{2}x+8\right)=0\)
\(\Leftrightarrow\left(x^2+5x+8\right)\left(x^2+6x+8\right)=0\)
\(\Leftrightarrow\left(x^2+5x+8\right)\left(x+2\right)\left(x+4\right)=0\Rightarrow\left[{}\begin{matrix}x=-2\\x=-4\end{matrix}\right.\)