a:
\(M=\dfrac{x^4+2}{x^6+1}+\dfrac{x^2-1}{x^4-x^2+1}-\dfrac{x^2+3}{x^4+4x^2+3}\)
\(=\dfrac{x^4+2}{\left(x^2+1\right)\left(x^4-x^2+1\right)}+\dfrac{x^2-1}{x^4-x^2+1}-\dfrac{x^2+3}{\left(x^2+3\right)\cdot\left(x^2+1\right)}\)
\(=\dfrac{x^4+2+\left(x^2-1\right)\left(x^2+1\right)}{\left(x^2+1\right)\left(x^4-x^2+1\right)}-\dfrac{1}{x^2+1}\)
\(=\dfrac{2x^4+1-x^4+x^2-1}{\left(x^2+1\right)\left(x^4-x^2+1\right)}=\dfrac{x^4+x^2}{\left(x^2+1\right)\left(x^4-x^2+1\right)}\)
\(=\dfrac{x^2\left(x^2+1\right)}{\left(x^2+1\right)\left(x^4-x^2+1\right)}=\dfrac{x^2}{x^4-x^2+1}\)
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