If you write [tex]y=x^{1/3}[/tex], you'll notice this is really just a quadratic equation that can be factorized:
[tex]2x^{2/3}+3x^{1/3}-5=0\iff 2y^2+3y-5=0\iff (2y+5)(y-1)=0\iff(2x^{1/3}+5)(x^{1/3}-1)=0[/tex]
So you have
[tex]2x^{1/3}+5=0\implies 2x^{1/3}=-5\implies x^{1/3}=-\dfrac52\implies x=-\dfrac{125}8[/tex]
and
[tex]x^{1/3}-1=0\implies x^{1/3}=1\implies x=1[/tex]
