$$ \begin{aligned} \frac{1}{x(x+1)}+\frac{1}{(x+1)(x+2)}+\frac{1}{(x+2)(x+3)} &= 0&& \text{multiply ALL terms by } \color{blue}{ x(x+1)(x+2)(x+3) }. \\[1 em]x(x+1)(x+2)(x+3)\cdot\frac{1}{x(x+1)}+x(x+1)(x+2)(x+3)\cdot\frac{1}{(x+1)(x+2)}+x(x+1)(x+2)(x+3)\cdot\frac{1}{(x+2)(x+3)} &= x(x+1)(x+2)(x+3)\cdot0&& \text{cancel out the denominators} \\[1 em]x^4+7x^3+17x^2+17x+6+x^4+7x^3+16x^2+12x+x^4+7x^3+15x^2+9x &= 0&& \text{simplify left side} \\[1 em]2x^4+14x^3+33x^2+29x+6+x^4+7x^3+15x^2+9x &= 0&& \\[1 em]3x^4+21x^3+48x^2+38x+6 &= 0&& \\[1 em] \end{aligned} $$

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