jmc

If $2+\sqrt{2}$ is the root of a polynomial with rational coefficients, then so is $2-\sqrt{2}$. Their sum is $4$ and their product is $(2+\sqrt{2})(2-\sqrt{2})=4-2=2.$ Thus, the monic quadratic with roots $2+\sqrt{2}$ and $2-\sqrt{2}$ is $x^2-4x+2$.

If $1-\sqrt{3}$ is the root of a polynomial with rational coefficients, then so is $1+\sqrt{3}$. Their sum is $2$ and their product is $(1-\sqrt{3})(1+\sqrt{3})=1-3=-2.$ Thus, the monic quadratic with roots $1-\sqrt{3}$ and $1+\sqrt{3}$ is $x^2-2x-2$.

Thus, the monic quartic with roots $2+\sqrt{2}$ and $1-\sqrt{3}$ is $$(x^2-4x+2)(x^2-2x-2)=\boxed{x^4-6x^3+8x^2+4x-4}.$$