$(z + 1) (z - 1) (z + i) (z - i) = - i^{2} z^{2} + z^{4} + i^{2} - z^{2}$

Explanation

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$(z + 1) (z - 1) (z + i) (z - i) \overset{◯}{1} (1 z^{2} - z + z - 1) (z + i) (z - i) \overset{◯}{2} (1 z^{2} - 1) (z + i) (z - i) \overset{◯}{3} (1 z^{3} + i z^{2} - z - i) (z - i) \overset{◯}{4} \overset{◯}{5} - i^{2} z^{2} + z^{4} + i^{2} - z^{2}$
①	Multiply each term of $(z + 1)$ by each term in $(z - 1)$ . $(z + 1) \cdot (z - 1) = z^{2} - z + z - 1$
②	Combine like terms: $z^{2} - z + z - 1 = z^{2} - 1$
③	Multiply each term of $(z^{2} - 1)$ by each term in $(z + i)$ . $(z^{2} - 1) \cdot (z + i) = z^{3} + i z^{2} - z - i$
④	Multiply each term of $(z^{3} + i z^{2} - z - i)$ by each term in $(z - i)$ . $(z^{3} + i z^{2} - z - i) \cdot (z - i) = = z^{4} - i z^{3} + i z^{3} - i^{2} z^{2} - z^{2} + i z - i z + i^{2}$
⑤	Combine like terms: $z^{4} - i z^{3} + i z^{3} - i^{2} z^{2} - z^{2} + i z - i z + i^{2} = - i^{2} z^{2} + z^{4} + i^{2} - z^{2}$

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(z+1)(z−1)(z+i)(z−i)=−i2z2+z4+i2−z2

$(z + 1) (z - 1) (z + i) (z - i) = - i^{2} z^{2} + z^{4} + i^{2} - z^{2}$