Such a polynomial with rational coefficients is THIS


    g(x) =  = .


To make the coefficients INTEGER numbers, multiply it by 2.  You will get then another polynomial

    f(x) = (x+4)*(x+1)*x*(2x-3).     


which is just with INTEGER coefficients and is, therefore, your  ANSWER.


Next you can make FOIL to get the polynomial in a standard form.