<s>[INST] inequality involving the sides of a triangle I want to show that if $\alpha > \beta >\gamma> 0$ and $\beta - \gamma <\alpha < \beta+\gamma$   then   
$α^6-4α^4 β^2+5α^2 β^4-2β^6-4α^4γ^2+3α^2 β^2γ^2-β^4γ^2+5α^2γ^4-β^2γ^4-2γ^6<0$
I tried it in Mathematica with the Reduce command and it showed that it holds.
Can anyone help?
 [/INST] Note that
$$\alpha > \beta > \gamma > 0; \ \beta - \gamma < \alpha < \beta + \gamma
\quad \Longleftrightarrow \quad \alpha = r + 2s + t, \ \beta = r + s + t, \ \gamma = r + s; \ s, t, r > 0.$$
Denote LHS as $f$. Using the substitutions $\alpha = r + 2s + t, \ \beta = r + s + t, \ \gamma = r + s; \ s, t, r > 0$, we have
\begin{align}
f &= -20 r^4 s^2-20 r^4 s t-8 r^4 t^2-92 r^3 s^3-138 r^3 s^2 t-62 r^3 s t^2-8 r^3 t^3-149 r^2 s^4-298 r^2 s^3 t\\
&\quad -183 r^2 s^2 t^2-34 r^2 s t^3-2 r^2 t^4-96 r s^5-240 r s^4 t-200 r s^3 t^2-60 r s^2 t^3-4 r s t^4-18 s^6\\
&\quad -54 s^5 t-60 s^4 t^2-30 s^3 t^3-6 s^2 t^4.
\end{align}
Clearly $f < 0$.
</s>