Given that $A = (\sqrt{2008}+\sqrt{2009}),$ $B = (-\sqrt{2008}-\sqrt{2009}),$ $C = (\sqrt{2008}-\sqrt{200}),$ and $D = (\sqrt{200}-\sqrt{2008}),$ find $ABCD.$
We see that $$(A-B)^2=((\sqrt{2008}+\sqrt{2009})-(-\sqrt{2008}-\sqrt{2009}))^2=(2\sqrt{2008})^2=4(2008).$$We note that the square root of the $2008$s in the answer choices are $2 \sqrt{4 \cdot 502} = 4 \sqrt{502}.$ The product of the last two terms in $(C-D)^2$ is $$((\sqrt{2008}-\sqrt{200})-(\sqrt{200}-\sqrt{2008}))(2 \sqrt{2008}) = -4(200-2008) = 4(2008).$$Therefore, $(A-B)^2=(C-D)^2$ so $ABCD = (A-B)(C-D) = \left(4 \sqrt{2008}\right) \left(-4 \sqrt{502}\right) = \boxed{-4(2008)}.$
To find $ABCD,$ we need to find the value of $A \cdot B \cdot C \cdot D.$
Step 1: Simplify $A \cdot B.$
\[A \cdot B = (\sqrt{2008}+\sqrt{2009}) \cdot (-\sqrt{2008}-\sqrt{2009})\]
By the difference of squares, we have:
\[A \cdot B = - (\sqrt{2008})^2 - (\sqrt{2009})^2\]
\[A \cdot B = -2008 - 2009\]
\[A \cdot B = -4017\]
Step 2: Simplify $C \cdot D.$
\[C \cdot D = (\sqrt{2008}-\sqrt{2009}) \cdot (\sqrt{200}-\sqrt{2008})\]
By the difference of squares, we have:
\[C \cdot D = (\sqrt{2008})^2 - (\sqrt{2009})^2\]
\[C \cdot D = 2008 - 2009\]
\[C \cdot D = -1\]
Step 3: Find $ABCD.$
\[ABCD = A \cdot B \cdot C \cdot D = (-4017) \cdot (-1) = \boxed{4017}\]