Showing that complicated mixed polynomial is always positive I want to show that
$\left(132 q^3-175 q^4+73 q^5-\frac{39 q^6}{4}\right)+\left(-144 q^2+12 q^3+70 q^4-19 q^5\right) r+\left(80 q+200 q^2-243 q^3+100 q^4-\frac{31 q^5}{2}\right) r^2+\left(-208 q+116 q^2+24 q^3-13 q^4\right) r^3+\left(80-44 q-44 q^2+34 q^3-\frac{23 q^4}{4}\right) r^4$
is strictly positive whenever $q \in (0,1)$ (numerically, this holds for all $r \in \mathbb{R}$, although I'm only interested in $r \in (0,1)$).
Is that even possible analytically? Any idea towards a proof would be greatly appreciated. Many thanks!

EDIT: Here is some more information.
Let
$f(r) = A + Br + Cr^2 + Dr^3 + Er^4$
be the function as defined above.
Then it holds that
$f(r)$ is a strictly convex function in $r$ for $q \in (0,1)$, $f(0) > 0$, $f'(0) < 0$, and $f'(q) > 0$. Hence, for the relevant $q \in (0,1)$, $f(r)$ attains its minimum for  some $r^{min} \in (0,q)$.
$A$ is positive and strictly increasing in $q$ for the relevant $q \in (0,1)$,
$B$ is negative and strictly decreasing in $q$ for the relevant $q \in (0,1)$,
$C$ is positive and strictly increasing in $q$ for the relevant $q \in (0,1)$,
$D$ is negative and non-monotonic in $q$, and
$E$ is positive and strictly decreasing in $q$ for the relevant $q \in (0,1)$.

Because you want to show that this is always positive, consider what happens when $q$ and $r$ get really big. The polynomials with the largest powers will dominate the result.
You can solve this quite easily by approximating the final value using a large number of inequalities.
