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\left(x^{2}\right)^{2}-2x^{2}x^{-2}+\left(x^{-2}\right)^{2}+1
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x^{2}-x^{-2}\right)^{2}.
x^{4}-2x^{2}x^{-2}+\left(x^{-2}\right)^{2}+1
To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.
x^{4}-2+\left(x^{-2}\right)^{2}+1
Multiply x^{2} and x^{-2} to get 1.
x^{4}-2+x^{-4}+1
To raise a power to another power, multiply the exponents. Multiply -2 and 2 to get -4.
x^{4}-1+x^{-4}
Add -2 and 1 to get -1.
\left(x^{2}\right)^{2}-2x^{2}x^{-2}+\left(x^{-2}\right)^{2}+1
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x^{2}-x^{-2}\right)^{2}.
x^{4}-2x^{2}x^{-2}+\left(x^{-2}\right)^{2}+1
To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.
x^{4}-2+\left(x^{-2}\right)^{2}+1
Multiply x^{2} and x^{-2} to get 1.
x^{4}-2+x^{-4}+1
To raise a power to another power, multiply the exponents. Multiply -2 and 2 to get -4.
x^{4}-1+x^{-4}
Add -2 and 1 to get -1.