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Your second question is fairly easy to answer, though perhaps it's not a fun answer.

sage: type(a)
sage.rings.polynomial.multi_polynomial_element.MPolynomial_polydict

sage: solve(a,a)
TypeError: The first argument must be a symbolic expression or a list of symbolic expressions.

solve only works with symbolic expressions, not polynomials, which Sage definitely distinguishes between. (E.g., sin(a) for this a won't work.)