\(\left(x-1\right)^3+x^3+\left(x+1\right)^3=\left(x+2\right)^3\)
\(\Leftrightarrow x^3-3x^2+3x-1+x^3+x^3+3x^2+3x+1=x^3+6x^2+12x+8\) \(\Leftrightarrow3x^3+6x^2+6x=x^3+6x^2+12x+8\) \(\Leftrightarrow3x^3+6x^2+6x-x^3-6x^2-12x-8=0\)
\(\Leftrightarrow2x^2-6x-8=0\)
\(\Leftrightarrow2x^2-8x+2x-8=0\)
\(\Leftrightarrow\left(2x^2+2x\right)-\left(8x+8\right)=0\)
\(\Leftrightarrow2x\left(x+1\right)-8\left(x+1\right)=0\)
\(\Leftrightarrow\left(x+1\right)\left(2x-8\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x+1=0\\2x-8=0\end{matrix}\right.\left[{}\begin{matrix}x=-1\\x=4\end{matrix}\right.\)
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