Answer
$-1$
Work Step by Step
Factoring the expressions and then cancelling the common factor/s between the numerator and the denominator, then the given expression, $
\dfrac{3x-x^2}{x^3-27}\div\dfrac{x}{x^2+3x+9}
$, simplifies to
\begin{array}{l}\require{cancel}
\dfrac{3x-x^2}{x^3-27}\cdot\dfrac{x^2+3x+9}{x}
\\\\=
\dfrac{-x(-3+x)}{(x-3)(x^2+3x+9)}\cdot\dfrac{x^2+3x+9}{x}
\\\\=
\dfrac{-\cancel{x}(\cancel{x-3})}{(\cancel{x-3})(\cancel{x^2+3x+9})}\cdot\dfrac{\cancel{x^2+3x+9}}{\cancel{x}}
\\\\=
-1
.\end{array}
