Suppose that $P$ is the polynomial of least degree with integer coefficients such that $P(\sqrt{7}+\sqrt{5})=2(\sqrt{7}-\sqrt{5})$. Find $P(2)$.

We are given $P(\sqrt{7}+\sqrt{5})=2(\sqrt{7}-\sqrt{5})$.
Note that $(\sqrt{7}+\sqrt{5})(\sqrt{7}-\sqrt{5})=7-5=2$.
So $\sqrt{7}-\sqrt{5}=\frac{2}{\sqrt{7}+\sqrt{5}}$.
Thus, the relation is:

where $x=\sqrt{7}+\sqrt{5}$.
Since $P$ must be a polynomial with integer coefficients, we must clear the denominator. The minimal polynomial of $x=\sqrt{7}+\sqrt{5}$ is:

Thus $x(x^3-24x)=-4$, so $\frac{4}{x}=24x-x^3$.
Therefore, the polynomial of least degree with integer coefficients is:

We want to find $P(2)$: