First, simplify the expression (4a^3 + 5) + (4a^3 - 1 )^2 - 2(4a^3 + 5) ( 4a^3 - 1):
(4a^3 + 5) + (4a^3 - 1 )^2 - 2(4a^3 + 5) ( 4a^3 - 1)= 4a^3 + 5 + (4a^3 - 1)(4a^3 - 1) - 2(16a^6 - 1)= 4a^3 + 5 + (16a^6 - 8a^3 + 1) - 32a^6 + 2= -12a^3 - 26a^6 + 8
Next, simplify the expression (p - 2a) (p + 2a) - (p - a ) (p^2 + pa + a^2):
(p - 2a) (p + 2a) - (p - a ) (p^2 + pa + a^2)= p^2 + 2ap - 2ap - 4a^2 - p^3 - p^2a - pa^2 + p^2 + pa + a^2= -p^3 + 2p^2 + a^2 - pa + 2
Therefore, the final simplified expressions are:-12a^3 - 26a^6 + 8-p^3 + 2p^2 + a^2 - pa + 2
First, simplify the expression (4a^3 + 5) + (4a^3 - 1 )^2 - 2(4a^3 + 5) ( 4a^3 - 1):
(4a^3 + 5) + (4a^3 - 1 )^2 - 2(4a^3 + 5) ( 4a^3 - 1)
= 4a^3 + 5 + (4a^3 - 1)(4a^3 - 1) - 2(16a^6 - 1)
= 4a^3 + 5 + (16a^6 - 8a^3 + 1) - 32a^6 + 2
= -12a^3 - 26a^6 + 8
Next, simplify the expression (p - 2a) (p + 2a) - (p - a ) (p^2 + pa + a^2):
(p - 2a) (p + 2a) - (p - a ) (p^2 + pa + a^2)
= p^2 + 2ap - 2ap - 4a^2 - p^3 - p^2a - pa^2 + p^2 + pa + a^2
= -p^3 + 2p^2 + a^2 - pa + 2
Therefore, the final simplified expressions are:
-12a^3 - 26a^6 + 8
-p^3 + 2p^2 + a^2 - pa + 2