Let a real number such as $ | a | > $ 2. Prove that if $ a ^ {4} -4a ^ {2} + 2 $ and $ a ^ 5-5a ^ 3 + 5a $ are rational numbers, so $ a $ is a rational number too.

My attempt is the following.

$ a ^ 5-5a ^ 3 + 5a = a (a ^ {4} -4a ^ {2} +2-a ^ 2 + 3) $. Who can be written as $ a ( frac {c} {d} -a ^ 2 + 3) $ for some people $ c, d $ in the integers. Note, $ a ( frac {c} {d} -a ^ 2 + 3) = frac {g} {h} $ by our hypothesis. Now we have $ frac {ac-da ^ 3 + 3ad} {d} = frac {g} {h} $. I played with the expression but I can not seem to get the result.