Question

How do you factor completely `2x^3 + 4x^2 + 6x + 12`?

Answer 1

`(x^2+3)(2x+4)` or `(x+isqrt3)(x-isqrt3)(2x+4)`

Explanation:

`y=2x^3+4x^2+6x+12`

Moving `6x` closer to `2x^3`

`y=2x^3+6x+4x^2+12`

`y=2x(x^2+3)+4(x^2+3)`

Take out `x^2+3`

`y=(x^2+3)(2x+4)`

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we can continue factoring `x^2+3` using complex numbers: `i^2=-1`

`y=(x^2-(3i^2))(2x+4)`

Using: `a^2-b^2=(a+b)(a-b)`

`y=(x+isqrt3)(x-isqrt3)(2x+4)`

Answer 2

`2(x+2)(x+sqrt3i)(x-sqrt3i)`

Explanation:

`"take out "color(blue)"common factor of 2"`

`rArr2(x^3+2x^2+3x+6)`

`"when "x=-2tox^3+2x^2+3x+6=0`

`rArr(x+2)" is a factor"`

`color(red)(x^2)(x+2)cancel(color(magenta)(-2x^2))cancel(+2x^2)+3x+6`

`=color(red)(x^2)(x+2)color(red)(+3)(x+2)cancel(color(magenta)(-6))cancel(+6)`

`rArrx^3+2x^2+3x+6=(x+2)(x^2+3)`

`"factor "x^2+3`

`x^2+3=0rArrx^2=-3rArrx=+-sqrt3i`

`rArr2x^3+4x^2+6x+12`

`=2(x+2)(x+sqrt3i)(x-sqrt3i)`