Question
$$x^3+(1+x)^3-(2x+1)(x+1) = 0$$
Answer
$$ \begin{matrix}x_1 = 0 & x_2 = -\dfrac{ 1 }{ 2 } \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} x^3+(1+x)^3-(2x+1)(x+1) &= 0&& \text{simplify left side} \\[1 em]x^3+1+3x+3x^2+x^3-(2x+1)(x+1) &= 0&& \\[1 em]2x^3+3x^2+3x+1-(2x+1)(x+1) &= 0&& \\[1 em]2x^3+3x^2+3x+1-(2x^2+2x+x+1) &= 0&& \\[1 em]2x^3+3x^2+3x+1-(2x^2+3x+1) &= 0&& \\[1 em]2x^3+3x^2+3x+1-2x^2-3x-1 &= 0&& \\[1 em]2x^3+3x^2+3x+1-2x^2-3x-1 &= 0&& \\[1 em]2x^3+x^2 &= 0&& \\[1 em] \end{aligned} $$
In order to solve $ \color{blue}{ 2x^{3}+x^{2} = 0 } $, first we need to factor our $ x^2 $.
$$ 2x^{3}+x^{2} = x^2 \left( 2x+1 \right) $$$ x = 0 $ is a root of multiplicity $ 2 $.
The second root can be found by solving equation $ 2x+1 = 0$.
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