Try 1/Expand[1/frac].

On more general approaches:

In Mathematica, an expression (a b)/(a c) will auto-simplify to b/c, and there's not much that can be done about common factors that are factored out except to work around it in some way, such as HoldForm, Defer, Inactive or as in the OP, boxes. These are used to keep the division that leads to the auto-simplification from evaluating. Another way is to manipulate a b or a c so that a is not factored out. Common things to try are Expand[a b] and Simplify[a b]. Sometimes there is no way to avoid using the first kind of workaround, such as when the desired output is 1/(1/m).

Here's a step-by-step breakdown of how to multiply the top and bottom of a fraction by the same factor:

frac = -((8 m^2)/(3 + 10 m^2 + 3 m^4));

sgn = Simplify[Sign[frac], m \[Element] Reals && m != 0]; (* wrestle Mma over the sign *)
fac = sgn/Numerator[frac]; (* construct factor for multiplying *)
newnd = fac*NumeratorDenominator[frac]; (* mult. num. & den. by fac *)
newnd = Expand /@ newnd; (* un-factor the product(s) *)
Divide @@ newnd (* construct new fraction *)
(*  -(1/(5/4 + 3/(8 m^2) + (3 m^2)/8))  *)

Here's a quick way that goes along with the default treatment of signs:

1/Expand[Denominator[frac]/Numerator[frac]]
(*  1/(-(5/4) - 3/(8 m^2) - (3 m^2)/8)  *)

Note that Sign[] on an algebraic expression is usually difficult to resolve to a 1 or -1. The following, using an undocumented function, always resolves to 1 or -1 and often agrees with the user's idea of what the leading sign is:

sgn = (-1)^Boole[Internal`SyntacticNegativeQ[frac]]
(*  -1  *)

Thank you for the excellent solution, Michael! Your 1/Expand[1/frac] approach is really elegant. I have a follow-up question: I modified your step-by-step method to divide both numerator and denominator by the parameter part of the numerator (excluding the sign and coefficient), and I came up with the following code:

sgn = Simplify[
  Sign[frac], (var = Variables[Numerator@frac][[1]]) \[Element] 
    Reals && var != 0](*wrestle Mma over the sign*)
Sign@(Cases[Numerator[frac] , _?NumericQ, Infinity][[1]])
fac = sgn/((Numerator[
       frac]/((Sign@(Cases[Numerator[frac] , _?NumericQ, 
             Infinity][[1]]))*(Cases[Numerator[frac] , _?NumericQ, 
           Infinity][[1]]))) // 
    Expand)(*construct factor for multiplying*)
newnd = fac*NumeratorDenominator[frac]; (*mult.num.& den.by fac*)
newnd = Expand /@ newnd; (*un-factor the product(s)*)
Divide @@ (newnd // Expand )(*construct new fraction*)

It works, but the code feels quite verbose and the nesting is a bit hard to follow. I was wondering if there's a more concise or optimized way to achieve the same result.

This is one way to do it, but we must strip the formatting away if we need to reuse the result in further calculations:

in = -((8 m^2)/(3 + 10 m^2 + 3 m^4))
Numerator[in]
newForm = 
 in /. a_/b_ -> 
   DisplayForm[FractionBox[a/Numerator[in], (b/Numerator[in])]]
3*newForm
3*newForm /. {FractionBox -> Divide, 
   DisplayForm -> Identity} // Expand

I think it's equivalent to dividing by 1, which Mathematica simplifies:

(n/n)/(d/n) -> 1/(d/n) -> n/d