Zeros of a function of degree 4 I'm trying to show that the following function has no zeros
$$
60x^4-44x^3-25x^2-44x+60=0.
$$
I already tried using Eisenstein's criterium, but since the first and the last coefficient are both $60$, there is no prime which holds.

In this answer I showed that the polynomial
$$
p(x) = x^4 + ax^3 + bx^2 + ax + 1
$$
has no real zeros if $|a| \leq 4$ and $b > 2 |a| -  2$.
We have
$$
60x^4-44x^3-25x^2-44x+60 = 60 \left(x^4 - \frac{44}{60}x^3 - \frac{25}{60}x^2 - \frac{44}{60}x+1\right),
$$
so in this case $a = -44/60$ and $b = -25/60$.  Indeed, $|a| \leq 4$ and
$$
b = -\frac{25}{60} > -\frac{32}{60} = 2|a|-2,
$$
so the given polynomial has no real zeros.
