<s>[INST] Calculate the sum of all irrational roots of $4\sqrt[3]{8x- 3}- 8x^{3}- 3= 0$ 
Calculate the sum of all irrational roots of
$$4\sqrt[3]{8x- 3}- 8x^{3}- 3= 0$$

I'm not even sure how to begin here, I tried raising it to the power of three, tried writing $8x^{3}+ 3$ with $x^{3}+ y^{3}= \left ( x+ y \right )\left ( x^{2}- xy+ y^{2} \right ),$ but have had no meaningful results.
 [/INST] 
Problem. Half-solve
$$4\sqrt[3]{8x- 3}- 8x^{3}- 3= 0$$

Substitute $y^{3}= 8x- 3$ in the original equation_
$$\left ( 4y- 8x^{3}- 3 \right )+ \left ( y^{3}- 8x+ 3 \right )= \left ( y- 2x \right )\left ( 4x^{2}+ 2xy+ y^{2}+ 4 \right )= 0$$
With $y= 2x,$ we have the equivalent one
$$8x^{3}- x+ 3=\!\left ( 2x- 1 \right )\!\left ( 4x^{2}+ 2x- 3 \right )=\!0\!\Rightarrow\exists_{x_{1}, x_{2}\in\left ( {\mathbb{Q}}' \right )^{2}}x_{1}+ x_{2}=\!-\frac{1}{2}$$
With $4x^{2}+ 2xy+ y^{2}+ 4= 0,$ there are no irrational roots, because
$$4x^{2}+ 2xy+ y^{2}+ 4= y^{6}+ 4y^{4}+ 6y^{3}+ 16y^{2}+ 12y+ 73= 0\;{\rm illegal}$$
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