$$ \begin{aligned} (x+3)\frac{x+6}{x+3}(x+2) &= \frac{9}{x+6}&& \text{multiply ALL terms by } \color{blue}{ (x+3)(x+6) }. \\[1 em](x+3)(x+6)(x+3)\frac{x+6}{x+3}(x+2) &= (x+3)(x+6)\cdot\frac{9}{x+6}&& \text{cancel out the denominators} \\[1 em]x^4+17x^3+102x^2+252x+216 &= 9x+27&& \text{move all terms to the left hand side } \\[1 em]x^4+17x^3+102x^2+252x+216-9x-27 &= 0&& \text{simplify left side} \\[1 em]x^4+17x^3+102x^2+243x+189 &= 0&& \\[1 em] \end{aligned} $$

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