<s>[INST] Solve $\frac{5}{x+3}+\frac{2}{2x+6}=4$ I want to solve $\frac{5}{x+3}+\frac{2}{2x+6}=4$ for $x$
I try this;
$5(2x+6)+2(x+3)=4(x+3)(2x+6)$
$12x+36 = 4(2x^2+12x+18)$
$8x^2+36x+36=0$
Where I would then go to the formula, however the answer says -1.5, what am I doing wrong.
 [/INST] The quadratic equation you have obtained has two solutions: $-3$ and $-\frac{3}{2}$. The latter one is the correct answer. The former one is incorrect because it would lead to division by $0$ when we plug it into the original equation.
However, the problem need not be solved via quadratic equation. Observe that
$\frac{5}{x + 3} + \color{red}{\frac{2}{2x + 6}} = \frac{5}{x + 3} + \color{red}{\frac{1}{x + 3}} = \frac{6}{x + 3}$. Then the equation could be written as $4(x + 3) = 6$, so $x = \frac{6}{4} - 3 = -\frac{3}{2}$.
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