Answer
$-\displaystyle \frac{4x}{4x^{2}+1} $
Work Step by Step
To add rational expressions with the same denominator,
add numerators and place the sum over the common denominator.
If possible, factor and simplify the result.
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$\displaystyle \frac{x^{2}+9x}{4x^{2}-11x-3}+\frac{3x-5x^{2}}{4x^{2}-11x-3} = \displaystyle \frac{x^{2}+9x+3x-5x^{2}}{4x^{2}-11x-3}$
$= \displaystyle \frac{12x-4x^{2}}{4x^{2}-11x-3}$
... To factor the numerator, factor out -4x (to have x positive in the other factor)
... To factor the denominator, search for
... two factors of $ac=-12$ whose sum is $b=-11$
... we find $-12$ and $+1.$
... Rewrite $bx$ and factor in pairs:
$4x^{2}-11x-3$=$4x^{2}-12x+x-3$
$=4x(x-3)+(x-3)$
$=(x-3)(4x^{2}+1)$
$= \displaystyle \frac{-4x(x-3)}{(4x^{2}+1)(x-3)}\qquad$... reduce the common factors
$=-\displaystyle \frac{4x}{4x^{2}+1} $
