Calculus Allowed (But It Won't Be Pretty)

Geometry Level 4

If the maximum value of \[ \sqrt{x^{4} -3x^{2}+ 6x+ 13} - \sqrt{x^{4} -x^{2}+ 1} \] is of the form \( \sqrt{\alpha} \) for all real \(x\), then find \(\alpha\).

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