Solve (a-3)^2(a+3)^2-(a^2-9)^2-1

$$( a - 3 ) ^ { 2 } ( a + 3 ) ^ { 2 } - ( a ^ { 2 } - 9 ) ^ { 2 } - 1$$

Evaluate

$$-1$$

Solution Steps

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

$$\left(a^{2}-6a+9\right)\left(a+3\right)^{2}-\left(a^{2}-9\right)^{2}-1$$

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

$$\left(a^{2}-6a+9\right)\left(a^{2}+6a+9\right)-\left(a^{2}-9\right)^{2}-1$$

Use the distributive property to multiply a^{2}-6a+9 by a^{2}+6a+9 and combine like terms.

$$a^{4}-18a^{2}+81-\left(a^{2}-9\right)^{2}-1$$

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(a^{2}-9\right)^{2}.

$$a^{4}-18a^{2}+81-\left(\left(a^{2}\right)^{2}-18a^{2}+81\right)-1$$

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

$$a^{4}-18a^{2}+81-\left(a^{4}-18a^{2}+81\right)-1$$

To find the opposite of a^{4}-18a^{2}+81, find the opposite of each term.

$$a^{4}-18a^{2}+81-a^{4}+18a^{2}-81-1$$

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

$$-18a^{2}+81+18a^{2}-81-1$$

Combine -18a^{2} and 18a^{2} to get 0.

$$81-81-1$$

Subtract 81 from 81 to get 0.

$$0-1$$

Subtract 1 from 0 to get -1.

$$-1$$

Factor

$$-1$$