Find all real solutions of the equation $x^{10} - x^8 + 8x^6 - 24x^4 + 32x^2 - 48 = 0$ I have been able to factorize the polynomial as follows: $$(x^2 - 2)(x^8 + x^6 + 10x^4 - 4x^2 + 24)$$ from which $\sqrt2$ and $-\sqrt2$ are obvious solutions. My guess is that $x^8 + x^6 + 10x^4 - 4x^2 + 24 = 0$ does not have any real solutions, but how would I show that?

$x^8 + x^6 + 10x^4 - 4x^2 + 24\geq 10x^4 - 4x^2>0$ if $x^{2}>\frac 2 5$ and $x^8 + x^6 + 10x^4 - 4x^2 + 24\geq x^8 + x^6 + 10x^4 - \frac 8 5 + 24\geq 24 -\frac  8 5 >0$ if $x^{2}<\frac 2 5$.
