Factorize each of the following expressions:$x^3 + 6x^2 + 12x + 16$

Given:

$x^3 + 6x^2 + 12x + 16$

To do:

We have to factorize the given expression.

Solution:

We know that,

$a^3 + b^3 = (a + b) (a^2 - ab + b^2)$

$a^3 - b^3 = (a - b) (a^2 + ab + b^2)$

Therefore,

$x^3 + 6x^2 + 12x + 16 = (x)^3 + 3(x^2)(2) + 3(x)(4) + (2)^3 + 8$         [Since $a^3 + 3a^2b + 3ab^2 +b^3 = (a + b)^3$]

$= (x + 2)^3 + 8$

$= (x + 2)^3 + (2)^3$

$= (x + 2 + 2) [(x + 2)^2 - (x + 2) \times 2 + (2)^2]$

$= (x + 4) (x^2 + 4x + 4 - 2x - 4 + 4)$

$= (x + 4) (x^2 + 2x + 4)$

Hence, $x^3 + 6x^2 + 12x + 16 = (x + 4) (x^2 + 2x + 4)$.