Solve 5left(a^2-1right)left(a^2-2right)-left(3a^2-2right)^2+left(5+2{a}^2right)left(5-2{a}^2right)

$$5 \left( { a }^{ 2 } -1 \right) \left( { a }^{ 2 } -2 \right) - { \left(3 { a }^{ 2 } -2 \right) }^{ 2 } + \left( 5+2 { a }^{ 2 } \right) \left( 5-2 { a }^{ 2 } \right)$$

Evaluate

$$31-3a^{2}-8a^{4}$$

Solution Steps

Use the distributive property to multiply 5 by a^{2}-1.

$$\left(5a^{2}-5\right)\left(a^{2}-2\right)-\left(3a^{2}-2\right)^{2}+\left(5+2a^{2}\right)\left(5-2a^{2}\right)$$

Use the distributive property to multiply 5a^{2}-5 by a^{2}-2 and combine like terms.

$$5a^{4}-15a^{2}+10-\left(3a^{2}-2\right)^{2}+\left(5+2a^{2}\right)\left(5-2a^{2}\right)$$

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(3a^{2}-2\right)^{2}.

$$5a^{4}-15a^{2}+10-\left(9\left(a^{2}\right)^{2}-12a^{2}+4\right)+\left(5+2a^{2}\right)\left(5-2a^{2}\right)$$

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

$$5a^{4}-15a^{2}+10-\left(9a^{4}-12a^{2}+4\right)+\left(5+2a^{2}\right)\left(5-2a^{2}\right)$$

To find the opposite of 9a^{4}-12a^{2}+4, find the opposite of each term.

$$5a^{4}-15a^{2}+10-9a^{4}+12a^{2}-4+\left(5+2a^{2}\right)\left(5-2a^{2}\right)$$

Combine 5a^{4} and -9a^{4} to get -4a^{4}.

$$-4a^{4}-15a^{2}+10+12a^{2}-4+\left(5+2a^{2}\right)\left(5-2a^{2}\right)$$

Combine -15a^{2} and 12a^{2} to get -3a^{2}.

$$-4a^{4}-3a^{2}+10-4+\left(5+2a^{2}\right)\left(5-2a^{2}\right)$$

Subtract 4 from 10 to get 6.

$$-4a^{4}-3a^{2}+6+\left(5+2a^{2}\right)\left(5-2a^{2}\right)$$

Consider \left(5+2a^{2}\right)\left(5-2a^{2}\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 5.

$$-4a^{4}-3a^{2}+6+25-\left(2a^{2}\right)^{2}$$

Expand \left(2a^{2}\right)^{2}.

$$-4a^{4}-3a^{2}+6+25-2^{2}\left(a^{2}\right)^{2}$$

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

$$-4a^{4}-3a^{2}+6+25-2^{2}a^{4}$$

Calculate 2 to the power of 2 and get 4.

$$-4a^{4}-3a^{2}+6+25-4a^{4}$$

Add 6 and 25 to get 31.

$$-4a^{4}-3a^{2}+31-4a^{4}$$

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

$$-8a^{4}-3a^{2}+31$$

Expand

$$31-3a^{2}-8a^{4}$$

Solution Steps

Use the distributive property to multiply 5 by a^{2}-1.

$$\left(5a^{2}-5\right)\left(a^{2}-2\right)-\left(3a^{2}-2\right)^{2}+\left(5+2a^{2}\right)\left(5-2a^{2}\right)$$

Use the distributive property to multiply 5a^{2}-5 by a^{2}-2 and combine like terms.

$$5a^{4}-15a^{2}+10-\left(3a^{2}-2\right)^{2}+\left(5+2a^{2}\right)\left(5-2a^{2}\right)$$

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(3a^{2}-2\right)^{2}.

$$5a^{4}-15a^{2}+10-\left(9\left(a^{2}\right)^{2}-12a^{2}+4\right)+\left(5+2a^{2}\right)\left(5-2a^{2}\right)$$

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

$$5a^{4}-15a^{2}+10-\left(9a^{4}-12a^{2}+4\right)+\left(5+2a^{2}\right)\left(5-2a^{2}\right)$$

To find the opposite of 9a^{4}-12a^{2}+4, find the opposite of each term.

$$5a^{4}-15a^{2}+10-9a^{4}+12a^{2}-4+\left(5+2a^{2}\right)\left(5-2a^{2}\right)$$

Combine 5a^{4} and -9a^{4} to get -4a^{4}.

$$-4a^{4}-15a^{2}+10+12a^{2}-4+\left(5+2a^{2}\right)\left(5-2a^{2}\right)$$

Combine -15a^{2} and 12a^{2} to get -3a^{2}.

$$-4a^{4}-3a^{2}+10-4+\left(5+2a^{2}\right)\left(5-2a^{2}\right)$$

Subtract 4 from 10 to get 6.

$$-4a^{4}-3a^{2}+6+\left(5+2a^{2}\right)\left(5-2a^{2}\right)$$

Consider \left(5+2a^{2}\right)\left(5-2a^{2}\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 5.

$$-4a^{4}-3a^{2}+6+25-\left(2a^{2}\right)^{2}$$

Expand \left(2a^{2}\right)^{2}.

$$-4a^{4}-3a^{2}+6+25-2^{2}\left(a^{2}\right)^{2}$$

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

$$-4a^{4}-3a^{2}+6+25-2^{2}a^{4}$$

Calculate 2 to the power of 2 and get 4.

$$-4a^{4}-3a^{2}+6+25-4a^{4}$$

Add 6 and 25 to get 31.

$$-4a^{4}-3a^{2}+31-4a^{4}$$

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

$$-8a^{4}-3a^{2}+31$$