1. **State the problem:** Solve the equation $$2x^4 + 6x^2 = 8x^8$$ for $x$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero: $$2x^4 + 6x^2 - 8x^8 = 0$$ 3. **Factor out the greatest common factor (GCF):** The GCF is $2x^2$: $$2x^2\left( x^2 + 3 - 4x^6 \right) = 0$$ 4. **Set each factor equal to zero:** - First factor: $$2x^2 = 0$$ - Second factor: $$x^2 + 3 - 4x^6 = 0$$ 5. **Solve the first factor:** $$2x^2 = 0 \implies x^2 = 0 \implies x = 0$$ 6. **Solve the second factor:** Rewrite: $$x^2 + 3 - 4x^6 = 0 \implies -4x^6 + x^2 + 3 = 0$$ Multiply both sides by $-1$ for clarity: $$4x^6 - x^2 - 3 = 0$$ 7. **Substitute $y = x^2$ to reduce degree:** $$4y^3 - y - 3 = 0$$ 8. **Solve the cubic equation $4y^3 - y - 3 = 0$:** Try rational roots using factors of 3 over factors of 4: possible roots $\pm1, \pm\frac{3}{4}, \pm3, \pm\frac{1}{2}$. Test $y=1$: $$4(1)^3 - 1 - 3 = 4 - 1 - 3 = 0$$ So $y=1$ is a root. 9. **Factor out $(y-1)$:** Divide $4y^3 - y - 3$ by $(y-1)$: $$4y^3 - y - 3 = (y-1)(4y^2 + 4y + 3)$$ 10. **Solve quadratic $4y^2 + 4y + 3 = 0$:** Calculate discriminant: $$\Delta = 4^2 - 4 \times 4 \times 3 = 16 - 48 = -32 < 0$$ No real roots here. 11. **Real solutions for $y$ are $y=1$ only.** Recall $y = x^2$, so: $$x^2 = 1 \implies x = \pm 1$$ 12. **Summary of solutions:** $$x = 0, \pm 1$$ **Final answer:** $$\boxed{x = -1, 0, 1}$$