The problem is simpler than you thought. You more or less got the right answer, though! Congratulations!
Your line 2 should have been 1+(6x-10)^(1/3)=+ or - sqrt(3). In the rest of the solution sqrt(3) should be replaced by + or - sqrt(3) to produce two solutions.
6x-10=(sqrt(3)-1)^3 (your line 4) = 3sqrt(3)-3(sqrt(3))^2+3sqrt(3)-1 (binomial expansion) = 6sqrt(3)-10, because sqrt(3) squared is 3.
The -10s cancel out and we're left with:
6x=6sqrt(3), so x=sqrt(3)=1.732... (your line 6)
Your answer fluctuated at the 14th decimal place!
The other solution is x=-sqrt(3).