Answer
$\dfrac{1}{x+2}$
Work Step by Step
Adding the numerators and copying the denominator, the given expression, $
\dfrac{x-6}{x^2+5x+6}+\dfrac{9}{x^2+5x+6}
,$ simplifies to
\begin{array}{l}\require{cancel}
\dfrac{x-6+9}{x^2+5x+6}
\\\\=
\dfrac{x+3}{x^2+5x+6}
\\\\=
\dfrac{x+3}{(x+3)(x+2)}
\\\\=
\dfrac{\cancel{x+3}}{(\cancel{x+3})(x+2)}
\\\\=
\dfrac{1}{x+2}
.\end{array}
