As with every other GCF/LCM problem, your best approach MUST include a complete factorization.

y^{3}-4y = y�(y^{2}-4) = y�(y+2)�(y-2)

y^{4}-8y = y�(y^{3}-8) = y�(y-2)�(y^{2}+2y+4)

The LCM is the collection of factors that JUST contains each factor of each element. No need to count a factor from one element if it is already included in a previous element.

Just copy the first one:

y�(y+2)�(y-2)

Then add whatever it does not have, that the other does.

y^{skip this one}�(y-2)^{skip this one}�(y^{2}+2y+4)^{use this one}

y�(y+2)�(y-2)�(y^{2}+2y+4)

Gives the LCM

GCF is the collection of factors that are common to all elements.