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

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

Evaluate

$$4$$

Solution Steps

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

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

Use the distributive property to multiply a^{3}+a-a^{2}-1 by a+1 and combine like terms.

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

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

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

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

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

To find the opposite of a^{4}-2a^{2}+1, find the opposite of each term.

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

Combine a^{4} and -a^{4} to get 0.

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

Subtract 1 from -1 to get -2.

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

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

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

Combine 2a^{2} and -2a^{2} to get 0.

$$-2+6$$

Add -2 and 6 to get 4.

$$4$$

Factor

$$2^{2}$$