I am pretty sure early European mathematicians did treat polynomials as areas or volumes, x can be just a rectangle x units long in one dimension and 1 in the other, x^2 is a square, etc. This meant they had to go through contortions to avoid negative coefficients since they made no geometric sense. If a coefficient would otherwise be negative it would have to move to the other side of the equality, and be solved using a different method. Instead of a single quadratic formula they needed several different cases depending the exact form of the polynomial.