$$(x+1)(x^2-x+1)-2x = x(x+1)(x-1)$$

$$ \begin{aligned} (x+1)(x^2-x+1)-2x &= x(x+1)(x-1)&& \text{simplify left and right hand side} \\[1 em]x^3-x^2+x+x^2-x+1-2x &= (x^2+x)(x-1)&& \\[1 em]x^3-x^2+x+x^2-x+1-2x &= x^3-x^2+x^2-x&& \\[1 em]x^3-2x+1 &= x^3-x^2+x^2-x&& \\[1 em]x^3-2x+1 &= x^3-x&& \text{move all terms to the left hand side } \\[1 em]x^3-2x+1-x^3+x &= 0&& \text{simplify left side} \\[1 em]x^3-2x+1-x^3+x &= 0&& \\[1 em]-x+1 &= 0&& \text{ move the constants to the right } \\[1 em]-x &= -1&& \text{switch signs on both sides} \\[1 em]x &= 1&& \\[1 em] \end{aligned} $$

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