\(\dfrac{1}{x^2+2x-3}=\dfrac{1}{\left(x+1\right)^2}+\dfrac{1}{48}\left(x\ne-3;\pm1\right)\)
\(\Leftrightarrow\dfrac{1}{\left(x-1\right)\left(x+3\right)}=\dfrac{1}{\left(x+1\right)^2}+\dfrac{1}{48}\)
\(\Leftrightarrow\dfrac{48\left(x+1\right)^2}{48\left(x+1\right)^2\left(x-1\right)\left(x+3\right)}=\dfrac{48\left(x-1\right)\left(x+3\right)}{48\left(x+1\right)^2\left(x-1\right)\left(x+3\right)}+\dfrac{\left(x+1\right)^2\left(x-1\right)\left(x+3\right)}{48\left(x+1\right)^2\left(x-1\right)\left(x+3\right)}\)
\(\Leftrightarrow48\left(x^2+2x+1\right)=48\left(x^2+2x-3\right)+\left(x+1\right)^2\left(x-1\right)\left(x+3\right)\)
\(\Leftrightarrow48x^2+96x+48=48x^2+96x-144+\left(x^4+4x^3+2x^2-4x-3\right)\)
\(\Leftrightarrow48+144=x^4+4x^3+2x^2-4x-3\)
\(\Leftrightarrow x^4+4x^3+2x^2-4x-3-192=0\)
\(\Leftrightarrow x^4+4x^3+2x^2-4x-195=0\)
\(\Leftrightarrow x^4+7x^3+23x^2+65x-3x^3-21x-69x-195=0\)
\(\Leftrightarrow x\left(x^3+7x^3+23x+65\right)-3\left(x^3+7x^3+23x+65\right)=0\)
\(\Leftrightarrow\left(x-3\right)\left(x^3+7x^3+23x+65\right)=0\)
\(\Leftrightarrow\left(x-3\right)\left(x^3+5x^2+2x^2+10x+13x+65\right)=0\)
\(\Leftrightarrow\left(x-3\right)\left[x^2\left(x+5\right)+2x\left(x+5\right)+13\left(x+5\right)\right]=0\)
\(\Leftrightarrow\left(x-3\right)\left(x+5\right)\left(x^2+2x+13\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x-3=0\\x+5=0\\x^2+2x+13=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=3\\x=-5\end{matrix}\right.\left(tm\right)\) (vì `x^2+2x+13=0` vô nghiệm)
ĐKXĐ:...
Đặt \(\left(x+1\right)^2=t\)
\(\Rightarrow\dfrac{1}{t-4}=\dfrac{1}{t}+\dfrac{1}{48}\)
\(\Rightarrow48t=48\left(t-4\right)+t\left(t-4\right)\)
\(\Rightarrow...\)
