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As long as you are able to express your giant coefficients in Sage form, that should be both pretty trivial and pretty pointless.

Pretty trivial:

sage: var("a,b,c,d,e")
(a, b, c, d, e)
sage: E=a*x^4+b*x^3+c*x^2+d*x+e==0
sage: %time S=solve(E,x)
CPU times: user 4.7 s, sys: 273 ms, total: 4.97 s
Wall time: 4.31 s
sage: len(S)

You can then S2=S.subs({a:whatever, b:you, c:want, d:to, e:substitute}) to get your expressions.

Pretty pointless: with coonstant coefficuients, this explicit solution of the quartic doesn't \LaTeX on a single A4 portrait page. Unless your giant coefficients have very specialshape, simplifying "miraculously" the expressions of the roots, your result will be unusable pe se.

However, they can be manipulated by Sage for further processing. Wouldn't it be easier ti denote the (unexpressed, but knon existing) roots of your quartic by variables, and use that in the "further processing" ?