solve(x^3+a*x+2>0)

In Maple 2017.2:

solve(x^3+x*a+2 > 0, [a, x]);
           [[a = -3, x < 1, -2 < x], [a = -3, 1 < x]]

solve(x^4+x*a+1 > 0, [a, x]);
Error, (in rootbound) 1st argument must be a polynomial with numeric coefficients

The first one clearly doesn't give the complete set of solutions. It can be made to work by adding conditions on a such as a>0.

I'm not sure if symbolic operations take index=real[i] into account or just silently ignore it:

limit(RootOf(_Z^3+a*_Z+2, index = real[1]), a = -infinity);
                               0

In fact the root grows as sqrt at -infinity.