6x2+12x+10

x2+2x+2

=

6(x2+2x+2)−2

x2+2x+2

=6−

2

x2+2x+2

=6−

2

(x+1)2+1

∵（x+1）²≥0，
∴（x+1）²+1≥1，
即

1

(x+1)2+1

≤1，−

2

(x+1)2+1

≥−2，6−

2

(x+1)2+1

≥6−2=4，
∴

6x2+12x+10

x2+2x+2

可取的最小值为4．
故选A．