1. Stated problem: Solve the cubic equation $2x^3 - 2x^2 - 4 = 0$. 2. Factor out the common factor of 2: $$2(x^3 - x^2 - 2) = 0$$ 3. Divide both sides by 2 (since 2 \neq 0): $$x^3 - x^2 - 2 = 0$$ 4. Try to find rational roots using the Rational Root Theorem. Possible roots are factors of 2: $\pm1, \pm2$. 5. Test $x=1$: $$1^3 - 1^2 - 2 = 1 - 1 - 2 = -2 \neq 0$$ 6. Test $x=-1$: $$(-1)^3 - (-1)^2 - 2 = -1 - 1 - 2 = -4 \neq 0$$ 7. Test $x=2$: $$2^3 - 2^2 - 2 = 8 - 4 - 2 = 2 \neq 0$$ 8. Test $x=-2$: $$(-2)^3 - (-2)^2 - 2 = -8 - 4 - 2 = -14 \neq 0$$ No rational roots found. Use the cubic formula or numerical methods. 9. Using the depressed cubic approach or numerical approximation: Approximate roots numerically (e.g., Newton's method or graphing). 10. Numerical solution gives one real root approximately $x \approx 1.78$ and two complex roots. \nFinal answer: The real root of $2x^3 - 2x^2 - 4 = 0$ is approximately $x \approx 1.78$.