Solve (3)(a^2-1)^2-(a^2+1)(a^2+2)

$$( 3 ) ( a ^ { 2 } - 1 ) ^ { 2 } - ( a ^ { 2 } + 1 ) ( a ^ { 2 } + 2 )$$

Evaluate

$$2a^{4}-9a^{2}+1$$

Solution Steps

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(a^{2}-1\right)^{2}.

$$3\left(\left(a^{2}\right)^{2}-2a^{2}+1\right)-\left(a^{2}+1\right)\left(a^{2}+2\right)$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$3\left(a^{4}-2a^{2}+1\right)-\left(a^{2}+1\right)\left(a^{2}+2\right)$$

Use the distributive property to multiply 3 by a^{4}-2a^{2}+1.

$$3a^{4}-6a^{2}+3-\left(a^{2}+1\right)\left(a^{2}+2\right)$$

Use the distributive property to multiply a^{2}+1 by a^{2}+2 and combine like terms.

$$3a^{4}-6a^{2}+3-\left(a^{4}+3a^{2}+2\right)$$

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

$$3a^{4}-6a^{2}+3-a^{4}-3a^{2}-2$$

Combine 3a^{4} and -a^{4} to get 2a^{4}.

$$2a^{4}-6a^{2}+3-3a^{2}-2$$

Combine -6a^{2} and -3a^{2} to get -9a^{2}.

$$2a^{4}-9a^{2}+3-2$$

Subtract 2 from 3 to get 1.

$$2a^{4}-9a^{2}+1$$

Expand

$$2a^{4}-9a^{2}+1$$

Solution Steps

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(a^{2}-1\right)^{2}.

$$3\left(\left(a^{2}\right)^{2}-2a^{2}+1\right)-\left(a^{2}+1\right)\left(a^{2}+2\right)$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$3\left(a^{4}-2a^{2}+1\right)-\left(a^{2}+1\right)\left(a^{2}+2\right)$$

Use the distributive property to multiply 3 by a^{4}-2a^{2}+1.

$$3a^{4}-6a^{2}+3-\left(a^{2}+1\right)\left(a^{2}+2\right)$$

Use the distributive property to multiply a^{2}+1 by a^{2}+2 and combine like terms.

$$3a^{4}-6a^{2}+3-\left(a^{4}+3a^{2}+2\right)$$

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

$$3a^{4}-6a^{2}+3-a^{4}-3a^{2}-2$$

Combine 3a^{4} and -a^{4} to get 2a^{4}.

$$2a^{4}-6a^{2}+3-3a^{2}-2$$

Combine -6a^{2} and -3a^{2} to get -9a^{2}.

$$2a^{4}-9a^{2}+3-2$$

Subtract 2 from 3 to get 1.

$$2a^{4}-9a^{2}+1$$