<s>[INST] Show quartic polynomial has no real solutions To show a lower bound for the runtime of an algorithm, I need to show that 
$$
3 x^4 - \frac{64}{5} x^3 + \frac{192}{5} x^2 - \frac{192}{5} x+ 12   > 0 
$$
for all real numbers $x\in \mathbb{R}$.
I followed the Wikipedia instructions and checked the paper they are based on. 
With computer help it is easy to find that the discriminant is positive. Now comes the part that has me lost: 
Let $a,b,c,d,e$ be the coefficients. Then, according to both the paper and Wikipedia, the quartic function has real roots if both 
$$8 a c - 3 b^2 <0\text{ and }64 a^3 e - 16 a^2 c^2 + 16 a b^2 c - 16 a^2 b d - 3 b^4 <0$$
and it has no real roots if both are positive. However, the case where
$$8 a c - 3 b^2 >0 \text{ and } 64 a^3 e - 16 a^2 c^2 + 16 a b^2 c - 16 a^2 b d - 3 b^4 <0$$ is not accounted for. Unfortunately, this is the case for my polynomial.
I appreciate any help, including hints for writing such a proof for a paper.
 [/INST] this equal to:
$15x^4-6x^3+192x^2-192x+60>0$
$15x^4-6x^3+192x^2-192x+60=(3x^2-8.702x+25.0905)(5x^2-6.83x+2.3709)+.5112$
$3x^2-8.702x+25.0905>0,5x^2-6.83x+2.3709>0 \implies \\15x^4-6x^3+192x^2-192x+60>0.5 >0$
$25.0905=\dfrac{14439589717}{57550000},2.3709=\dfrac{272893639}{11510000}$
</s>