How can we prove that the following inequality in four real variables? $$3dc^2+3bc^2+3d^2c6dc+3b^2c4abc6bc+3ad^24abd+3a^2d6ad+ab^2+a^2b2ab\ge 0~,$$ with $a,b,c,d\ge 2$.
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$\begingroup$ Could you please suggest me a helpful software tool to verify that this inequality is true? WolframAlpha is not helpful in this case. $\endgroup$ – Penelope Benenati Jul 19 '20 at 17:02

$\begingroup$ Why do you believe it to be true? $\endgroup$ – LSpice Jul 19 '20 at 17:10

$\begingroup$ Yes, I run several computation experiments and there are some other reason to believe that is true, related to the original problem where this inequality has been originated. $\endgroup$ – Penelope Benenati Jul 19 '20 at 17:11

1$\begingroup$ i think this can be proven using amgm and the fact that all the variables are greater than 2 $\endgroup$ – vidyarthi Jul 19 '20 at 17:14

3$\begingroup$ Do you know when does equality occur? Did you try the substitutions $a=2+x$ etc? $\endgroup$ – Fedor Petrov Jul 19 '20 at 17:34

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The inequality is false for $a=1799, b=105, c=1024, d=4$.
This counterexample was found and verified with Mathematica, as follows:

4$\begingroup$ Thank you Iosif. How did you find these values? $\endgroup$ – Penelope Benenati Jul 19 '20 at 18:14

4$\begingroup$ @PenelopeBenenati : I have added the way the counterexample was found. $\endgroup$ – Iosif Pinelis Jul 19 '20 at 18:49
