This system was solved in May 2020, using solveSystem in Macaulay2 v1.15 with an Intel(R) Core(TM) i5-5250U CPU at 1.60GHz.
There were 32 solutions found in 0.0944 seconds (with a Bezout bound of 32).
Reference: "Computation of all solutions to a system of polynomial equations" by M. Kojima and S. Mizuno (pages 131-157), "Finding all solutions to a system of polynomial equations" by A.H. Wright (pages 125-133), and "A simple homotopy method for determining all isolated solutions to polynomial systems" by W. Zulehner (pages 167-177).
See also: http://homepages.math.uic.edu/~jan/Demo/wright.html.
i1 : wright(QQ)
2 2 2
o1 = {x - x + x + x + x + x - 10, x + x - x + x + x + x - 10, x
1 1 2 3 4 5 2 1 2 3 4 5 3
------------------------------------------------------------------------
2 2
+ x + x - x + x + x - 10, x + x + x + x - x + x - 10, x + x
1 2 3 4 5 4 1 2 3 4 5 5 1
------------------------------------------------------------------------
+ x + x + x - x - 10}
2 3 4 5
o1 : List
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The object wright is a method function.