For polynomials with real coefficients, complex roots "z" always go in pairs 

(z, z-conjugated).


The conjugated to  4i  is the complex number  -4i.


So, your polynomial is  a*(x-3)*(x-4i)*(x+4i) = .


The leading coefficient "a" is an arbitrary real number, or, 
if you really are seeking for rational coefficients, then "a" must be a rational number.