$$\frac{10}{x}+3 = \frac{x+9}{x-4}$$
Answer
$$ \begin{matrix}x_1 = -\dfrac{ 5 }{ 2 } & x_2 = 8 \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{10}{x}+3 &= \frac{x+9}{x-4}&& \text{multiply ALL terms by } \color{blue}{ x(x-4) }. \\[1 em]x(x-4)\cdot\frac{10}{x}+x(x-4)\cdot3 &= x(x-4)\frac{x+9}{x-4}&& \text{cancel out the denominators} \\[1 em]10x-40+3x^2-12x &= x^2+9x&& \text{simplify left side} \\[1 em]3x^2-2x-40 &= x^2+9x&& \text{move all terms to the left hand side } \\[1 em]3x^2-2x-40-x^2-9x &= 0&& \text{simplify left side} \\[1 em]2x^2-11x-40 &= 0&& \\[1 em] \end{aligned} $$
$ 2x^{2}-11x-40 = 0 $ is a quadratic equation.
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