3(1+a²+a^4)-(1+a+a²)²
=3a^4+3a²+3-a^4-a²-1-2a³-2a²-2a
=2a^4-2a³-2a+2
=2a³(a-1)-2(a-1)
=2(a-1)(a³-1)
=2(a-1)²(a²+a+1)
a²+a+1=a²+a+1/4+3/4=(a+1/2)²+3/4>0
(a-1)²>=0
所以
2(a-1)²(a²+a+1)>=0
所以3(1+a²+a^4)>=(1+a+a²)²
当a=1时取等号