In order to cover also the multiplicities of the solutions of an algebraic equation, one has the following chances:
$(1)$ Use solve with the option multiplicities=True:
var('x');
f = x^5 - x^4 - 2*x^3 + 2*x^2 + x - 1
solve(f==0, x, multiplicities=True)
This gives:
([x == -1, x == 1], [2, 3])
indicating that the solutions $x=-1$ and respectively $x=1$ have multiplicities $2$, and respectively $3$.
To obtain this information, just ask for the doc string of the used function solve:
sage: ?solve
Signature: solve(f, *args, **kwds)
Docstring:
Algebraically solve an equation or system of equations (over the
complex numbers) for given variables. Inequalities and systems of
inequalities are also supported.
INPUT:
* "f" - equation or system of equations (given by a list or tuple)
* "*args" - variables to solve for.
* "solution_dict" - bool (default: False); if True or non-zero,
return a list of dictionaries containing the solutions. If there
are no solutions, return an empty list (rather than a list
containing an empty dictionary). Likewise, if there's only a
single solution, return a list containing one dictionary with
that solution.
There are a few optional keywords if you are trying to solve a
single equation. They may only be used in that context.
* "multiplicities" - bool (default: False); if True, return
corresponding multiplicities. This keyword is incompatible with
"to_poly_solve=True" and does not make any sense when solving
inequalities.
and many other lines.
$(2)$ Use an algebraic object, thus making the used polynomial element in a polynomial ring,
then ask for the roots of the polynomial.
R.<x> = PolynomialRing(QQ)
f = x^5 - x^4 - 2*x^3 + 2*x^2 + x - 1
f.roots()
This gives:
[(-1, 2), (1, 3)]
a list of tuples of the shape ( solution, its multiplicity ). So
the solution $x=-1$ has multiplicity $2$, and the solution $x=1$ has multiplicity $3$.
This time, if the multiplicities are not wanted, we can ask for this other version of the result...
sage: f.roots(multiplicities=False)
[-1, 1]
