This might work for you:
And[Reduce[(-4 x^2 + 8 x + 12)/(-x - 1) < 0, x] ,
ReleaseHold[
WolframAlpha[
"Domain (-4 x^2+8 x+12)/(-x-1)", {{"Result", 1},
"Output"}]]] // Simplify
On the Raspberry Pi with the post Mathematica 9 release this command is more elegant and does not need to invoke WA:
And[Reduce[(-4 x^2 + 8 x + 12)/(-x - 1) < 0, x] ,
FunctionDomain[(-4 x^2 + 8 x + 12)/(-x - 1), x]] // Simplify
because on the Raspberry we have the command FunctionDomain, which is new. I believe that Reduce does not consider the domain of the underlying function: that's not a bug, that's a feature.
This one might even be a bit nicer:
Reduce[{(-4 x^2 + 8 x + 12)/(-x - 1) < 0 &&
FunctionDomain[(-4 x^2 + 8 x + 12)/(-x - 1), x]}, x] // Simplify
If you have the Raspberry, you can of course put this together into one function:
advReduce[eqn_] :=
Reduce[{eqn && FunctionDomain[eqn[[1, 1]], x]}, x] // Simplify
You can call that function with the equation/inequality as an argument:
advReduce[(-4 x^2 + 8 x + 12)/(-x - 1) < 0]
(* x < -1 || -1 < x < 3 *)
You can also write a similar function in Mathematica 9 which then uses Wolfram Alpha:
advReduce2[eqn_] :=
Reduce[
eqn && ReleaseHold[
WolframAlpha[
"Domain" <> (eqn[[1]] // InputForm // ToString), {{"Result",
1}, "Output"}]]] // Simplify
With the function call
advReduce2[(-4 x^2 + 8 x + 12)/(-x - 1) < 0]
which gives:
x < -1 || -1 < x < 3
Cheers,
M.