(a) Determine if the upper bound theorem identifies 4 as an upper bound for the real zeros of .
(b) Determine if the lower bound theorem identifies -6 as a lower bound for the real zeros of .

To check if 4 is an upper bound for the real zeros of , you can evaluate and then check the synthetic division of by . If the synthetic division gives you all non-negative values in the remainder, then 4 is indeed an upper bound.

For evaluated at 4, you would find that all coefficients in the final row of the synthetic division remain non-negative, thus confirming 4 as an upper bound.

Now, to determine if -6 is a lower bound for the real zeros of , use the same method: substitute -6 into . If the coefficients during the synthetic division by turn out to be alternating in sign starting from a positive value, then -6 serves as a lower bound for the real zeros. Alternatively, check if is non-positive.