ĐKXĐ: \(x\ne\left\{-2;-4;-6;-8\right\}\)
\(\dfrac{\left(x+2\right)^2+2}{x+2}+\dfrac{\left(x+8\right)^2+8}{x+8}=\dfrac{\left(x+4\right)^2+4}{x+4}+\dfrac{\left(x+6\right)^2+6}{x+6}\)
\(\Leftrightarrow x+2+\dfrac{2}{x+2}+x+8+\dfrac{8}{x+8}=x+4+\dfrac{4}{x+4}+x+6+\dfrac{6}{x+6}\)
\(\Leftrightarrow\dfrac{2}{x+2}+\dfrac{8}{x+8}=\dfrac{4}{x+4}+\dfrac{6}{x+6}\)
\(\Leftrightarrow\dfrac{2}{x+2}-\dfrac{4}{x+4}+\dfrac{8}{x+8}-\dfrac{6}{x+6}=0\)
\(\Leftrightarrow\dfrac{-2x}{\left(x+2\right)\left(x+4\right)}+\dfrac{2x}{\left(x+6\right)\left(x+8\right)}=0\)
\(\Rightarrow\left[{}\begin{matrix}x=0\\\left(x+2\right)\left(x+4\right)=\left(x+6\right)\left(x+8\right)\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}x=0\\x^2+6x+8=x^2+14x+48\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}x=0\\x=-5\end{matrix}\right.\)
