\(A=\frac{\left(x^2+1\right)^2+4}{\left(x^2+1\right)^2}=1+\frac{4}{\left(x^2+1\right)^2}\)
\(x^2\ge0\Rightarrow x^2+1\ge1\Rightarrow\left(x^2+1\right)^2\ge1\Rightarrow\frac{4}{\left(x^2+1\right)^2}\le4\Rightarrow1+\frac{4}{\left(x^2+1\right)^2}\le5\)
\(M\text{ax}A=5\Leftrightarrow x=0\)