$$ \begin{aligned} \frac{1}{x(x+1)}+\frac{1}{(x+1)(x+2)}+\frac{1}{(x+2)(x+3)} &= (2x+1)x&& \text{multiply ALL terms by } \color{blue}{ x(x+1)(x+2)(x+3) }. \\[1 em]x(x+1)(x+2)(x+3)\cdot\frac{1}{x(x+1)}+x(x+1)(x+2)(x+3)\cdot\frac{1}{(x+1)(x+2)}+x(x+1)(x+2)(x+3)\cdot\frac{1}{(x+2)(x+3)} &= x(x+1)(x+2)(x+3)(2x+1)x&& \text{cancel out the denominators} \\[1 em]x^4+7x^3+17x^2+17x+6+x^4+7x^3+16x^2+12x+x^4+7x^3+15x^2+9x &= 2x^6+13x^5+28x^4+23x^3+6x^2&& \text{simplify left side} \\[1 em]2x^4+14x^3+33x^2+29x+6+x^4+7x^3+15x^2+9x &= 2x^6+13x^5+28x^4+23x^3+6x^2&& \\[1 em]3x^4+21x^3+48x^2+38x+6 &= 2x^6+13x^5+28x^4+23x^3+6x^2&& \text{move all terms to the left hand side } \\[1 em]3x^4+21x^3+48x^2+38x+6-2x^6-13x^5-28x^4-23x^3-6x^2 &= 0&& \text{simplify left side} \\[1 em]-2x^6-13x^5-25x^4-2x^3+42x^2+38x+6 &= 0&& \\[1 em] \end{aligned} $$

This page was created using