Let's analyze the given equation:

$2a^2b^2 + 2b^2c^2 + 2c^2a^2 = a^4b^4c^4$

We'll first check if the equation holds by simplifying each side.

Step 1: Simplifying the left-hand side (LHS)

$\text{LHS} = 2a^2b^2 + 2b^2c^2 + 2c^2a^2$

Factor out the common factor of $2$: $\text{LHS} = 2(a^2b^2 + b^2c^2 + c^2a^2)$

This is the simplified form of the left-hand side. Now, let's check the right-hand side.

Step 2: Right-hand side (RHS)

$\text{RHS} = a^4b^4c^4$