\(A=\dfrac{x^2+2x+1}{x^2-4x+5}=\dfrac{\left(x+1\right)^2}{\left(x-2\right)^2+1}\)
Do \(\left\{{}\begin{matrix}\left(x+1\right)^2\ge0\\\left(x-2\right)^2+1>0\end{matrix}\right.\) ;\(\forall x\)
\(\Rightarrow A\ge0\) ;\(\forall x\)
\(A_{min}=0\) khi \(x=-1\)