jmc

We start by constructing a quadratic polynomial with $\sqrt{2}+\sqrt{3}$ and $\sqrt{2}-\sqrt{3}$ as roots. The sum of the roots is $\sqrt{2}+\sqrt{3}+\sqrt{2}-\sqrt{3}=2\sqrt{2}.$ The product of the roots is $(\sqrt{2}+\sqrt{3})(\sqrt{2}-\sqrt{3})=2-3=-1.$ Thus a quadratic with the roots $\sqrt{2}+\sqrt{3}$ and $\sqrt{2}-\sqrt{3}$ is $$x^2-2\sqrt{2}x-1.$$Next, we want to get rid of the irrational coefficients. We can write $x^2-2\sqrt{2}x-1$ as $x^2-1-2\sqrt{2}x$. Then, multiplying by $x^2-1+2\sqrt{2}x$ gives us $$(x^2-1-2\sqrt{2}x)(x^2-1+2\sqrt{2}x)=(x^2-1{)}^2-(2\sqrt{2}x{)}^2=\boxed{x^4-10x^2+1}$$which is a monic polynomial of degree $4$ with rational coefficients that has $\sqrt{2}+\sqrt{3}$ as a root.