FWIW, since regular languages (aka rational languages) are closed under complementation, it's always possible to find an regular expression (aka rational expression) that negates another expression. But not many tools implement this.
Vcsn supports this operator (which it denotes {c}, postfix).
You first define the type of your expressions: labels are letter (lal_char) to pick from a to z for instance (defining the alphabet when working with complementation is, of course, very important), and the "value" computed for each word is just a Boolean: true the word is accepted, false, rejected.
In Python:
In [5]: import vcsn
c = vcsn.context('lal_char(a-z), b')
c
Out[5]: {a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z} → 𝔹
then you enter your expression:
In [6]: e = c.expression('(hede){c}'); e
Out[6]: (hede)^c
convert this expression to an automaton:
In [7]: a = e.automaton(); a

finally, convert this automaton back to a simple expression.
In [8]: print(a.expression())
\e+h(\e+e(\e+d))+([^h]+h([^e]+e([^d]+d([^e]+e[^]))))[^]*
where + is usually denoted |, \e denotes the empty word, and [^] is usually written . (any character). So, with a bit of rewriting ()|h(ed?)?|([^h]|h([^e]|e([^d]|d([^e]|e.)))).*.
You can see this example here, and try Vcsn online there.
([^h]*(h([^e]|$)|he([^d]|$)|hed([^e]|$)))*? The idea is simple. Keep matching until you see the start of the unwanted string, then only match in the N-1 cases where the string is unfinished (where N is the length of the string). These N-1 cases are "h followed by non-e", "he followed by non-d", and "hed followed by non-e". If you managed to pass these N-1 cases, you successfully didn't match the unwanted string so you can start looking for[^h]*again – stevendesu Sep 29 '11 at 3:44^([^h]*(h([^e]|$)|he([^d]|$)|hed([^e]|$))?)*$this fails when instances of "hede" are preceded by partial instances of "hede" such as in "hhede". – jaytea Sep 10 '12 at 10:41