Solve (3a^2+2a+8)^2-(a^2-2a-8)^2

$$( 3 a ^ { 2 } + 2 a + 8 ) ^ { 2 } - ( a ^ { 2 } - 2 a - 8 ) ^ { 2 }$$

Evaluate

$$8a^{2}\left(a^{2}+2a+8\right)$$

Solution Steps

Square 3a^{2}+2a+8.

$$9a^{4}+12a^{3}+52a^{2}+32a+64-\left(a^{2}-2a-8\right)^{2}$$

Square a^{2}-2a-8.

$$9a^{4}+12a^{3}+52a^{2}+32a+64-\left(a^{4}-4a^{3}-12a^{2}+32a+64\right)$$

To find the opposite of a^{4}-4a^{3}-12a^{2}+32a+64, find the opposite of each term.

$$9a^{4}+12a^{3}+52a^{2}+32a+64-a^{4}+4a^{3}+12a^{2}-32a-64$$

Combine 9a^{4} and -a^{4} to get 8a^{4}.

$$8a^{4}+12a^{3}+52a^{2}+32a+64+4a^{3}+12a^{2}-32a-64$$

Combine 12a^{3} and 4a^{3} to get 16a^{3}.

$$8a^{4}+16a^{3}+52a^{2}+32a+64+12a^{2}-32a-64$$

Combine 52a^{2} and 12a^{2} to get 64a^{2}.

$$8a^{4}+16a^{3}+64a^{2}+32a+64-32a-64$$

Combine 32a and -32a to get 0.

$$8a^{4}+16a^{3}+64a^{2}+64-64$$

Subtract 64 from 64 to get 0.

$$8a^{4}+16a^{3}+64a^{2}$$

Expand

$$8a^{4}+16a^{3}+64a^{2}$$

Solution Steps

Square 3a^{2}+2a+8.

$$9a^{4}+12a^{3}+52a^{2}+32a+64-\left(a^{2}-2a-8\right)^{2}$$

Square a^{2}-2a-8.

$$9a^{4}+12a^{3}+52a^{2}+32a+64-\left(a^{4}-4a^{3}-12a^{2}+32a+64\right)$$

To find the opposite of a^{4}-4a^{3}-12a^{2}+32a+64, find the opposite of each term.

$$9a^{4}+12a^{3}+52a^{2}+32a+64-a^{4}+4a^{3}+12a^{2}-32a-64$$

Combine 9a^{4} and -a^{4} to get 8a^{4}.

$$8a^{4}+12a^{3}+52a^{2}+32a+64+4a^{3}+12a^{2}-32a-64$$

Combine 12a^{3} and 4a^{3} to get 16a^{3}.

$$8a^{4}+16a^{3}+52a^{2}+32a+64+12a^{2}-32a-64$$

Combine 52a^{2} and 12a^{2} to get 64a^{2}.

$$8a^{4}+16a^{3}+64a^{2}+32a+64-32a-64$$

Combine 32a and -32a to get 0.

$$8a^{4}+16a^{3}+64a^{2}+64-64$$

Subtract 64 from 64 to get 0.

$$8a^{4}+16a^{3}+64a^{2}$$