Many ODEs do not have symbolic solutions in terms of commonly defined functions. The ODE in the question probably falls in that category. However, it should be noted that DSolve sometimes is unable to solve ODEs that do have symbolic solutions in terms of commonly defined functions.
We can solve numerically:
sol = With[{m = 3, n = 2},
NDSolve[{y''''[x] + 2 y[x]^m + y[x]^n == x*Exp[-x], y[0] == 1,
y'[0] == 1, y''[0] == 0, y'''[0] == 0}, y, {x, 0, 2}]]
Plot[y[x] /. sol, {x, 0, 2}, PlotRange -> All]
Regards M.I.
