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TLDR: I want to solve function $(-4x^2-6x+4)/(x^2+1)^2$ for $0$. How can I get the polynomial out of the denominator so I can apply the quadratic formula?

Long form: I'm trying to find the horizontal tangents of the function $f(x)=(4x+3)/(x^2+1)$, I'm pretty sure I got the first derivative correct and I want to set the result of the derivative to $0$ so I can find all the line formulae where the slope is $0$.

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up vote 6 down vote accepted

Simply note that $ (x^2+1)^2 \geq 1 $ and in particular, $ (x^2+1)^2 \neq 0 .$ Thus $$ \frac{-4x^2-6x+4}{(x^2+1)^2} = 0 $$ if and only if $$ -4x^2-6x+4 = 0 .$$

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