Step 1: Simplify the expression under the first square root.
Consider $ \sin^4 x + 4\cos^2 x $. Replace $ \cos^2 x $ with $ 1 - \sin^2 x $: $ \sin^4 x + 4(1 - \sin^2 x) = \sin^4 x - 4\sin^2 x + 4 $.
Step 2: Recognise this as a perfect square.
$ \sin^4 x - 4\sin^2 x + 4 = (\sin^2 x - 2)^2 $. So the first square root is $ \sqrt{(\sin^2 x - 2)^2} = |\sin^2 x - 2| $. Since $ \sin^2 x \leq 1 < 2 $, we have $ \sin^2 x - 2 < 0 $, so $ |\sin^2 x - 2| = 2 - \sin^2 x $.
Step 3: Simplify the expression under the second square root.
Consider $ \cos^4 x + 4\sin^2 x $. Replace $ \sin^2 x $ with $ 1 - \cos^2 x $: $ \cos^4 x + 4(1 - \cos^2 x) = \cos^4 x - 4\cos^2 x + 4 = (\cos^2 x - 2)^2 $. So the second square root equals $ |\cos^2 x - 2| = 2 - \cos^2 x $ (since $ \cos^2 x \leq 1 < 2 $).
Step 4: Compute the difference.
\[ (2 - \sin^2 x) - (2 - \cos^2 x) = \cos^2 x - \sin^2 x \]
Step 5: Apply the double angle identity.
The identity $ \cos 2x = \cos^2 x - \sin^2 x $ gives the answer directly.
Step 6: State the final answer.
\[ \boxed{\cos 2x} \]