Given,
$\begin{gathered}
\frac{1}{x^4+x^2+1}=-\frac{A x+B}{x^2+x+1}+\frac{C x+D}{x^2-x+1} \\
\Rightarrow 1=(A x+B)\left(x^2-x+1\right)+(C x+D)\left(x^2+x+1\right) \\
\Rightarrow 1=\left(A x^3+A x^{2+} A x+B x^2-B x+B\right) \\
\quad+\left(C x^3+C x^2+C x+D x^2+D x+D\right) \\
\Rightarrow 1=(A+C) x^3+(-A+B+C+D) x^2 \\
\quad+(A-B+C+D) x+(B+D)
\end{gathered}$
On comparing, the coefficient of like powers on both sides, we get
$\begin{aligned}
A+C & =0 \\
-A+B+C+D & =0 \\
A-B+C+D & =0 \\
B+D & =1
\end{aligned}$
and
On adding Eqs. (ii) and (iii), we get
$\begin{aligned}
2(C+D) & =0 \\
\Rightarrow \quad C+D & =0
\end{aligned}$