a.
\(M=\dfrac{x^4+2}{\left(x^2+1\right)\left(x^4-x^2+1\right)}+\dfrac{x^2-1}{x^4-x^2+1}-\dfrac{x^2+3}{\left(x^2+1\right)\left(x^2+3\right)}\)
\(=\dfrac{x^4+2}{\left(x^2+1\right)\left(x^4-x^2+1\right)}+\dfrac{x^2-1}{x^4-x^2+1}-\dfrac{1}{x^2+1}\)
\(=\dfrac{x^4+2}{\left(x^2+1\right)\left(x^4-x^2+1\right)}+\dfrac{\left(x^2-1\right)\left(x^2+1\right)}{\left(x^2+1\right)\left(x^4-x^2+1\right)}-\dfrac{x^4-x^2+1}{\left(x^2+1\right)\left(x^4-x^2+1\right)}\)
\(=\dfrac{x^4+2+x^4-1-x^4+x^2-1}{\left(x^2+1\right)\left(x^4-x^2+1\right)}\)
\(=\dfrac{x^2\left(x^2+1\right)}{\left(x^2+1\right)\left(x^4-x^2+1\right)}=\dfrac{x^2}{x^4-x^2+1}\)
b.
\(M=\dfrac{x^4-x^2+1-x^4+2x^2-1}{x^4-x^2+1}=1-\dfrac{\left(x^2-1\right)^2}{x^4-x^2+1}\le1\)
\(M_{max}=1\) khi \(x=\pm1\)
