I'm trying to determine if $$\bigl(x+y)^4(y+z)^4(z+x)^4 \geq$$ $$8x^2y^2z^2\bigl((x+y)^2 + (y+z)^2\bigr)\bigl((y+z)^2 + (z+x) ^2\bigr)\bigl((z+x)^2 + (x+y)^2\bigr)$$
for $x,y,z>0$.
I'm trying to determine if $$\bigl(x+y)^4(y+z)^4(z+x)^4 \geq$$ $$8x^2y^2z^2\bigl((x+y)^2 + (y+z)^2\bigr)\bigl((y+z)^2 + (z+x) ^2\bigr)\bigl((z+x)^2 + (x+y)^2\bigr)$$
for $x,y,z>0$.