\(\dfrac{x^2+2x+2}{x+1}>\dfrac{x^2+4x+5}{x+2}-1\left(x\ne-1,-2\right)\)
\(\Leftrightarrow\dfrac{x^2+2x+1+1}{x+1}>\dfrac{x^2+4x+4+1}{x+2}-1\)
\(\Leftrightarrow\dfrac{\left(x+1\right)^2+1}{x+1}>\dfrac{\left(x+2\right)^2+1}{x+2}-1\)\(\Leftrightarrow\dfrac{\left(x+1\right)^2}{x+1}+\dfrac{1}{x+1}>\dfrac{\left(x+2\right)^2}{x+2}+\dfrac{1}{x+2}-1\)
\(\Leftrightarrow x+1+\dfrac{1}{x+1}>x+2+\dfrac{1}{x+2}-1\)
\(\Leftrightarrow\dfrac{1}{x+1}>\dfrac{1}{x+2}\)
\(\Leftrightarrow\dfrac{1}{x+1}-\dfrac{1}{x+2}>0\)
\(\Leftrightarrow\dfrac{x+2-x-1}{\left(x+1\right)\left(x+2\right)}>0\)
\(\Leftrightarrow\dfrac{1}{\left(x+1\right)\left(x+2\right)}>0\)mà 1 > 0 \(\Rightarrow\left(x+1\right)\left(x+2\right)>0\)
\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x+1>0\\x+2>0\end{matrix}\right.\\\left\{{}\begin{matrix}x+1< 0\\x+2< 0\end{matrix}\right.\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x>-1\\x>-2\end{matrix}\right.\\\left\{{}\begin{matrix}x< -1\\x< -2\end{matrix}\right.\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x>-1\\x< -2\end{matrix}\right.\)
⇔x2+2x+1+1x+1>x2+4x+4+1x+2−1⇔x2+2x+1+1x+1>x2+4x+4+1x+2−1
⇔(x+1)2x+1+1x+1>(x+2)2x+2+1x+2−1⇔(x+1)2x+1+1x+1>(x+2)2x+2+1x+2−1
⇔1x+1>1x+2⇔1x+1>1x+2
⇔x+2−x−1(x+1)(x+2)>0⇔x+2−x−1(x+1)(x+2)>0