Question

How do you factor `2x^3+3x^2-16x-24`?

Answer 1

Given `2x^3+3x^2-16x-24`

Regroup as
`color(red)(2x^3-16x) + color(blue)(3x^2-24)`

Extract factor from each pair
`= color(red)((2x)(x^2-8)) + color(blue)((3)(x^2-8)`

Extract the `(x^2-8)` common factor
`=(2x+3)(x^2-8)`

If you are willing to employ irrational constants, this can be further factored (using the difference of squares) as
`=(2x-3)(x+2sqrt(2))(x-2sqrt(2))`

Answer 2

Notice that the ratio between the 1st and 2nd term is the same as that between the 3rd and 4th term. So grouping will work...

`2x^3+3x^2-16x-24 = (2x^3+3x^2)-(16x+24)`

`=x^2(2x+3)-8(2x+3)`

`=(x^2-8)(2x+3)`

`=(x-sqrt(8))(x+sqrt(8))(2x+3)`

`=(x-2sqrt(2))(x+2sqrt(2))(2x+3)`