Question
Factorize the following expressions: $x^3 + 6x^2 + 12x + 16$
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Answer
$=x^3+6 x^2+12 x+8+8 $
$=x^3+3 \times x^2 \times 2+3 \times x \times 2^2+2^3+8 $
$=(x+2)^3+8\left[\therefore a^3+3 a^2 b+3 a b^2+b^3=(a+b)^3\right] $
$=(x+2)^3+23 $
$=(x+2+2)\left((x+2)^2-2(x+2)+2^2\right)\left[\therefore a^3+b^3=(a+b)\left(a^2-a b+b^2\right)\right] $
$=(x+2+2)\left(x^2+4+4 x-2 x-4+4\right)\left[\therefore(a+b)^2=a^2+b^2+2 a b\right] $
$=(x+4)\left(x^2+4+2 x\right) $
$\therefore x^3+6 x^2+12 x+16=(x+4)\left(x^2+4+2 x\right)$
$=x^3+3 \times x^2 \times 2+3 \times x \times 2^2+2^3+8 $
$=(x+2)^3+8\left[\therefore a^3+3 a^2 b+3 a b^2+b^3=(a+b)^3\right] $
$=(x+2)^3+23 $
$=(x+2+2)\left((x+2)^2-2(x+2)+2^2\right)\left[\therefore a^3+b^3=(a+b)\left(a^2-a b+b^2\right)\right] $
$=(x+2+2)\left(x^2+4+4 x-2 x-4+4\right)\left[\therefore(a+b)^2=a^2+b^2+2 a b\right] $
$=(x+4)\left(x^2+4+2 x\right) $
$\therefore x^3+6 x^2+12 x+16=(x+4)\left(x^2+4+2 x\right)$
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