How to get access of a single solution from a bunch of solutions

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The first trick is to use the solution_dict parameter to get the solutions as Python dictionaries, instead of lists of expressions:

sage: relations = [f(0) == -2, f(2) == 0,  f1(4) == 0, f(4) == 8,  f2(2) == 0]
sage: sols = solve(relations, a, b, c, d, e, solution_dict=True)
sage: sols
[{a: 3/16, b: -41/16, c: 87/8, d: -12, e: -2}]

As you can see, sols is a list, with a single element, that is a dictionary:

sage: sols[0]
{a: 3/16, b: -41/16, c: 87/8, d: -12, e: -2}
sage: sols[0][a]
3/16
sage: sols[0][b]
-41/16

Now, you can use this dictionary to substitute the symbols a, b, c, d, e with their values:

sage: f.substitute(sols[0])
x |--> 3/16*x^4 - 41/16*x^3 + 87/8*x^2 - 12*x - 2