If $a, b, c, d> 0$ then I try to prove the following inequality.. If $a, b, c, d> 0$ then prove that $$a^2 b^2 c^2 + b^2 c^2 d^2 + a^2 b^2 d^2 + a^2 c^2 d^2 \geq ab^2 c^2 d + ab^2 cd^2 + a^2 bcd^2 + a^2bc^2d$$
I'm using AM GM relation but I'm not getting the answer, probably due to a conceptual mistake?

We may write $a^2b^2c^2 + b^2c^2d^2 + a^2b^2d^2 + a^2c^2d^2$ as
$$\frac{a^2b^2c^2 + b^2c^2d^2}{2} + \frac{b^2c^2d^2 + a^2b^2d^2}{2} + \frac{a^2b^2d^2 + a^2c^2d^2}{2} + \frac{a^2c^2d^2 + a^2b^2c^2}{2}$$
Apply the AM-GM inequality to each term to obtain the result.
