1. The problem is to determine if the equation $2x^4 + 6x^2 = 8x^8$ is true for all values of $x$. 2. To check this, we can try to simplify or rearrange the equation and analyze it. 3. Rewrite the equation: $$2x^4 + 6x^2 = 8x^8$$ 4. Move all terms to one side: $$2x^4 + 6x^2 - 8x^8 = 0$$ 5. Factor out the common term with the lowest power of $x$, which is $x^2$: $$x^2(2x^2 + 6 - 8x^6) = 0$$ 6. Simplify inside the parentheses: $$x^2(2x^2 + 6 - 8x^6) = x^2(6 + 2x^2 - 8x^6) = 0$$ 7. For the product to be zero, either: - $x^2 = 0$ which means $x=0$, or - $6 + 2x^2 - 8x^6 = 0$ 8. The second equation is a polynomial in $x^2$: Let $y = x^2$, then: $$6 + 2y - 8y^3 = 0$$ 9. Rearranged: $$-8y^3 + 2y + 6 = 0$$ 10. This cubic equation does not hold for all $y$, so the original equation is not true for all $x$. 11. Check a simple value, for example $x=1$: Left side: $2(1)^4 + 6(1)^2 = 2 + 6 = 8$ Right side: $8(1)^8 = 8$ So it holds for $x=1$. 12. Check $x=0$: Left side: $0 + 0 = 0$ Right side: $0$ True. 13. Check $x=2$: Left side: $2(2)^4 + 6(2)^2 = 2(16) + 6(4) = 32 + 24 = 56$ Right side: $8(2)^8 = 8(256) = 2048$ Not equal. 14. Therefore, the equation is not true for all $x$, only for specific values. Final answer: The equation $2x^4 + 6x^2 = 8x^8$ is not true for all $x$.