Find a non-zero polynomial $P$ with integer coefficients such that $P(\sqrt{2008}+\sqrt{2009})=0$.
Let $x=\sqrt{2008}+\sqrt{2009}$. We have: $x^{2}=2008+2009+2 \sqrt{2008 \times 2009}$. Therefore, $4 \times 2008 \times 2009=$ $\left(x^{2}-2008-2009\right)^{2}$. Hence, $x^{4}-8034 x^{2}+1=0$.