Which of the following is the factor of $4 a^{2} + b^{2} - 4 a b + 2 b - 4 a + 1$

a - b

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a + b - 2

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a - b + 2

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2 a - b - 1

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Solution

The correct option is D $2 a - b - 1$
$4 a^{2} + b^{2} - 4 a b + 2 b - 4 a + 1$
$= {(2 a)}^{2} - 2 (b) (2 a) - 2 (2 a) + (b^{2} + 2 b + 1)$
$= {(2 a)}^{2} - 2 (2 a) (b + 1) + {(b + 1)}^{2}$
$= {(2 a - (b + 1))}^{2}$
$= (2 a - b - 1) (2 a - b + 1)$

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