Factorize :
$a^{4} + 5 a^{2} + 9$

(a^{3} + a - 3) (a^{2} - a + 1)

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(a^{2} + a + 3) (a^{2} - a + 3)

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(a^{2} + a - 3) (a^{2} - a + 4)

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(a^{3} + a + 3) (a^{2} - a - 3)

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Solution

The correct option is B $(a^{2} + a + 3) (a^{2} - a + 3)$
$a^{4} + 5 a^{2} + 9$
Using,
$(a + b)^{2} = a^{2} + 2 a b + b^{2}$

$a^{4} + 5 a^{2} + 9$
$= (a^{2})^{2} + 2 (a^{2}) (3^{2}) + 3^{2} - a^{2}$
$= (a^{2} + 3)^{2} - a^{2}$

Using,
$(a^{2} - b^{2}) = (a + b) (a - b)$

$a^{4} + 5 a^{2} + 9$
$= (a^{2} + 3)^{2} - a^{2}$
$= (a^{2} + a + 3) (a^{2} + 3 - a)$
$= (a^{2} + a + 3) (a^{2} - a + 3)$

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