Solve (a^{2}-1)*(a^{2}+1)-(a^{2}-1)^{2}-(2a^2-1)^2+2a^2*(a^2-3)=

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

Evaluate

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

Solution Steps

Consider \left(a^{2}-1\right)\left(a^{2}+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

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

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

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

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

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

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

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

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

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

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

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

Subtract 1 from -1 to get -2.

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

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

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

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

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

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

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

Combine 2a^{2} and 4a^{2} to get 6a^{2}.

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

Subtract 1 from -2 to get -3.

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

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

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

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

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

Combine 6a^{2} and -6a^{2} to get 0.

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

Expand

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

Solution Steps

Consider \left(a^{2}-1\right)\left(a^{2}+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

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

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

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

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

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

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

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

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

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

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

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

Subtract 1 from -1 to get -2.

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

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

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

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

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

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

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

Combine 2a^{2} and 4a^{2} to get 6a^{2}.

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

Subtract 1 from -2 to get -3.

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

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

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

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

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

Combine 6a^{2} and -6a^{2} to get 0.

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