$$ \begin{aligned} \frac{8}{x+4}+\frac{28}{x-1} &= 0&& \text{multiply ALL terms by } \color{blue}{ (x+4)(x-1) }. \\[1 em](x+4)(x-1)\cdot\frac{8}{x+4}+(x+4)(x-1)\cdot\frac{28}{x-1} &= (x+4)(x-1)\cdot0&& \text{cancel out the denominators} \\[1 em]8x-8+28x+112 &= 0&& \text{simplify left side} \\[1 em]36x+104 &= 0&& \text{ move the constants to the right } \\[1 em]36x &= -104&& \text{ divide both sides by $ 36 $ } \\[1 em]x &= -\frac{104}{36}&& \\[1 em]x &= -\frac{26}{9}&& \\[1 em] \end{aligned} $$

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