Question 283066
Just looking at it I would suspect that (x-6) and/or (x+3) are factors of the denominators.
Answer :{{{-27/((x-3) (x+2) (x+6))}}}

or Answer:{{{ -27/(x^3+5x^2-12x-36) }}}

How did I get there?
Start with
{{{((x-6)/(x^2-x-6))-((x+3)/(x^2+8x+12))}}}
The long way:
multiply first fraction by {{{(6+x)/(6+x)}}} and the second fraction by {{{(x-3)/(x-3)}}} aka {{{(-3+x)/(-3+x)}}} to make a common denominator
  = {{{ ((x-6) (6+x))/((x^2-x-6) (6+x)) + (-(x+3) (-3+x))/((x^2+8x+12) (-3+x))}}} 
multiply it out  = {{{(-36+x^2)/(-36-12x+5 x^2+x^3)+(9-x^2)/(-36-12x+5x^2+x^3)}}}
collect terms  =  {{{(-36+x^2 + 9-x^2)/(-36-12x+5x^2+x^3) }}}
add ={{{-27/(-36-12 x+5 x^2+x^3) }}} 
rearrange
Answer:{{{ -27/(x^3+5x^2-12x-36) }}}
or
Answer :{{{-27/((x-3) (x+2) (x+6))}}}
Another way to to do the same thing:
The short way:
{{{x^2-x-6=(x-3)(x+2)}}} 
{{{x^2+8x+12=(x+6)(x+2)}}}

{{{((x-6)/(x^2-x-6))-((x+3)/(x^2+8x+12))}}}
{{{((x-6)/(x-3)(x+2))-((x+3)/(x+6)(x+2))}}}
multiply by {{{(6+x)/(6+x)}}}
multiply by {{{(x-3)/(x-3)}}}
{{{((6+x)/(6+x))*((x-6)/(x-3)(x+2))-((x-3)/(x-3))*((x+3)/(x+6)(x+2))}}}
multiply out the numerators
and get the same as above.
Answer :{{{-27/((x-3) (x+2) (x+6))}}}