jmc

Since the polynomial has rational coefficients, it must also have $1-\sqrt{2}$ and $1-\sqrt{3}$ as roots. Then, the polynomial must be divisible by the two polynomials $$(x-(1+\sqrt{2}))(x-(1-\sqrt{2}))=x^2-2x-1$$and $$(x-(1+\sqrt{3}))(x-(1-\sqrt{3}))=x^2-2x-2.$$It follows that the polynomial we seek is given by $$(x^2-2x-1)(x^2-2x-2)=\boxed{x^4-4x^3+x^2+6x+2}.$$