no of roots

P(x)=(x-a₁) (x-a₃) (x-a₅) + 3(x-a₂) (x-a₄) (x-a₆)

P(a₁)<0
P(a₂)>0

implies there's a root in (a₁,a₂)

again,
P(a₃)>0
P(a₄)<0
P(a₅)<0
P(a₆)>0

Therefore, there are 3 real roots in the intervals (a₁,a₂) , (a₃,a₄), (a₅,a₆).

Hence, only correct option (a). [1]

Can someone tell me, from which book this question has been taken?

I think its a variant of a past IIT-JEE question.