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Solve for x 2x^(2/3) - 5x^(1/3) - 3 = 0
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Prepare your equation to factoring using grouping-regrouping (in the same way as you did in your post)
2x^(2/3) - 6x^(1/3) + x^(1/3) - 3 = 0
Now factor left side step by step
2x^(1/3)*(x^(1/3)-3) + (x^(1/3) - 3) =
(x^(1/3)-3)*(2x^(1/3)+1) = 0.
From this factored form you have EITHER first parentheses expression is zero,
OR the second parentheses expression is zero.
(a) Case x^(1/3)-3 = 0.
It gives x^(1/3) = 3, which implies x = 3^3 = 27.
(b) Case 2x^(1/3)+1 = 0.
It gives 2x^(1/3) + 1 0, which implies x^(1/3) = , x = = .
Solved.
The answers/(the solutions) are x= 27 and/or x= , same as in your textbook.
Victory (!)
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In your post, your very first step was correct, but your factoring after that was wrong.