Sage cannot solve non-linear polynomial systems symbolically!

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You mean solutions involving square roots etc. ( like $\frac{1}{2} \left(2-i \sqrt{2 \left(\sqrt{33}-1\right)}\right)$ ) rather than floating point results, right? You should force the integers to be symbols. The following is a "impractical" workaround:

x, y, z = var('x y z')
var('a,b,c')
eq1 = x + y + z == a
eq2 = x^2 + 2*y + 3*z == b
eq3 = x + y + z^2 == c
soln=solve([eq1,eq2,eq3],x,y,z)
show(soln[0][0].rhs().subs(a=2,b=4,c=10))

Tip: in the Ask Sage search box, type: polynomial system.