1. **State the problem:** Solve for $x$ in the equation $$2x^2 - \frac{8}{4x} = 3x^2 + \frac{6x}{2x}.$$\n\n2. **Rewrite the equation clearly:** $$2x^2 - \frac{8}{4x} = 3x^2 + \frac{6x}{2x}.$$\n\n3. **Simplify the fractions:** \n- Simplify $\frac{8}{4x} = \frac{8}{4x} = \frac{2}{x}$.\n- Simplify $\frac{6x}{2x} = 3$ (since $x \neq 0$).\nSo the equation becomes $$2x^2 - \frac{2}{x} = 3x^2 + 3.$$\n\n4. **Bring all terms to one side:** $$2x^2 - \frac{2}{x} - 3x^2 - 3 = 0,$$ which simplifies to $$-x^2 - \frac{2}{x} - 3 = 0.$$\n\n5. **Multiply through by $x$ to clear the denominator (assuming $x \neq 0$):** $$x \cdot \left(-x^2 - \frac{2}{x} - 3\right) = 0 \implies -x^3 - 2 - 3x = 0.$$\n\n6. **Rewrite:** $$-x^3 - 3x - 2 = 0,$$ or equivalently $$x^3 + 3x + 2 = 0.$$\n\n7. **Solve the cubic equation $x^3 + 3x + 2 = 0$:**\nTry rational roots using factors of 2: $\pm1, \pm2$.\n- For $x = -1$: $$(-1)^3 + 3(-1) + 2 = -1 - 3 + 2 = -2 \neq 0.$$\n- For $x = -2$: $$(-2)^3 + 3(-2) + 2 = -8 - 6 + 2 = -12 \neq 0.$$\n- For $x = 1$: $$1 + 3 + 2 = 6 \neq 0.$$\n- For $x = 2$: $$8 + 6 + 2 = 16 \neq 0.$$\nNo rational roots found, so use the cubic formula or factor by depressed cubic method.\n\n8. **Use depressed cubic substitution:** The equation is $$x^3 + 3x + 2 = 0.$$\nDiscriminant $\Delta = \left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3$ where $p=3$, $q=2$.\nCalculate: $$\Delta = \left(\frac{2}{2}\right)^2 + \left(\frac{3}{3}\right)^3 = 1^2 + 1^3 = 1 + 1 = 2 > 0,$$ so one real root.\n\n9. **Find the real root using Cardano's formula:**\n$$x = \sqrt[3]{-\frac{q}{2} + \sqrt{\Delta}} + \sqrt[3]{-\frac{q}{2} - \sqrt{\Delta}} = \sqrt[3]{-1 + \sqrt{2}} + \sqrt[3]{-1 - \sqrt{2}}.$$\n\n10. **Approximate the root:**\nCalculate numerically: \n- $-1 + \sqrt{2} \approx -1 + 1.414 = 0.414$, cube root $\approx 0.75$.\n- $-1 - \sqrt{2} \approx -1 - 1.414 = -2.414$, cube root $\approx -1.35$.\nSum: $0.75 - 1.35 = -0.6$ approximately.\n\n**Final answer:** $$x \approx -0.6.$$