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\left(x^{4}-1\right)\left(x^{4}+1\right)-\left(x^{4}+1\right)^{2}+2\left(x^{4}+1\right)
Use the distributive property to multiply x^{2}-1 by x^{2}+1 and combine like terms.
\left(x^{4}\right)^{2}-1-\left(x^{4}+1\right)^{2}+2\left(x^{4}+1\right)
Consider \left(x^{4}-1\right)\left(x^{4}+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.
x^{8}-1-\left(x^{4}+1\right)^{2}+2\left(x^{4}+1\right)
To raise a power to another power, multiply the exponents. Multiply 4 and 2 to get 8.
x^{8}-1-\left(\left(x^{4}\right)^{2}+2x^{4}+1\right)+2\left(x^{4}+1\right)
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x^{4}+1\right)^{2}.
x^{8}-1-\left(x^{8}+2x^{4}+1\right)+2\left(x^{4}+1\right)
To raise a power to another power, multiply the exponents. Multiply 4 and 2 to get 8.
x^{8}-1-x^{8}-2x^{4}-1+2\left(x^{4}+1\right)
To find the opposite of x^{8}+2x^{4}+1, find the opposite of each term.
-1-2x^{4}-1+2\left(x^{4}+1\right)
Combine x^{8} and -x^{8} to get 0.
-2-2x^{4}+2\left(x^{4}+1\right)
Subtract 1 from -1 to get -2.
-2-2x^{4}+2x^{4}+2
Use the distributive property to multiply 2 by x^{4}+1.
-2+2
Combine -2x^{4} and 2x^{4} to get 0.
0
Add -2 and 2 to get 0.
\left(x^{4}+1\right)\left(\left(x^{2}-1\right)\left(x^{2}+1\right)-1-x^{4}+2\right)
Factor out common term x^{4}+1 by using distributive property.
0
Consider \left(x^{2}-1\right)\left(x^{2}+1\right)-1-x^{4}+2. Simplify.