Simplify the following and express it in the form $a + i b$ :
$(1 - i)^{2} (1 + i) - (3 - 4 i)^{2}$

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Solution

$(1 - i)^{2} (1 + i) - (3 - 4 i)^{2}$
$= (1 - 2 i + i^{2}) (1 + i) - (9 - 24 i + 16 i^{2})$
$= 1 - 2 i - 1 + i + 2 - i - 9 + 24 i + 16$
$= (9 + 22 i)$
Which is of the form $a + i b$ .

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