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A polynomial vanishes ift each of its coefficients vanish, so you can work with polynomials as follows:

sage: var('x_00, x_10, x_20')
(x_00, x_10, x_20)
sage: R.<x,y,z> = SR[]
sage: R
Multivariate Polynomial Ring in x, y, z over Symbolic Ring
sage: P = x^2*x_00 + x*x_10*y + x_20*y^2 - -2*x*y
sage: P
x_00*x^2 + (x_10 + 2)*x*y + x_20*y^2
sage: solve([p == 0 for p in P.coefficients()], [x_00 , x_10 , x_20])
[[x_00 == 0, x_10 == -2, x_20 == 0]]

For more algebraic tasks, instead of SR[x,y,z]It is also possible to work in the ring ZZ[x_00, x_10, x_20][x,y,z], but you get the idea.