I have the following inequality regarding convex pentagons: $$5(a^2+b^2+c^2+d^2+e^2+2(ab+bc+cd+de+ea)+2(ac+bd+ce+da+eb)\geq 18(a^2+b^2+c^2+d^2+e^2-(ea\cos{\alpha}+ab\cos{\beta}+bc\cos{\gamma}+cd\cos{\delta}+de\cos{\epsilon}))$$
In this answer to other question I proved that
$$\cos{\alpha}+\cos{\beta}+\cos{\gamma}+\cos{\delta}+\cos{\epsilon}\leq -\frac{1}{2}$$
I have checked with Wolfram Mathematica that this result implies that the inequality is false if all the variables are positive reals; however, I am not able to prove it. I will appreciate any help with it.
Thanks in advance!
EDIT
Working a bit more on the inequality, to prove that it is false, it suffices to show that $$-18(ea\cos{\alpha}+ab\cos{\beta}+bc\cos{\gamma}+cd\cos{\delta}+de\cos{\epsilon})> 7(ea+ab+bc+cd+de)$$ This is so because such a result would imply that $$5(a^2+b^2+c^2+d^2+e^2)+3(ab+bc+cd+de+ea)+10(ac+bd+ce+da+eb)+7(ab+bc+cd+de+ea)> 18(a^2+b^2+c^2+d^2+e^2)+7(ab+bc+cd+de+ea)$$ Which is false by the rearrangement inequality.