It is given that, f(x) is a quartic polynomial with integer coefficients and four integer roots.
Constant Term of f(x)=6
(a)
By rational root theorem , the roots of the polynomial can be

So, Yes x=3, can be root of f(x).
(b)
Yes, a quartic Polynomial , can have double root as x=3, that is two roots equal to , x=3.
For example,Consider the Polynomial

has two zeroes equal to 3.