Question

How do you factor `8x^3-12x^2+2x-3`?

Answer 1

`(8x^3-12x+2x-3=color(green)((2x-3)(4x^2+1))`

Explanation:

Notice that the ratio of the coefficients of the terms `8x^3` and `2x`: `8:2=4:1`
is the same as the ratio of the coefficients of the terms `-12x^2` and `-3`: `-12:-3 =4:1`

This hints hat we should group the original expression as
`color(white)("XXX")(color(red)(8x^3+2x))-(color(blue)(12x^2+3))`

`color(white)("XXX")=color(red)(2x(4x^2+1)))-color(blue)(3(4x^2+1))`

then extracting the common factor `(4x^2+1)`
`color(white)("XXX")=(2x-3)(4x^2+1)`

(Note that since `4x^2 >= 0` for `AAx in RR`,
`color(white)("XXX")4x^2+1` can not be equal to `0`
`color(white)("XXX")`and `(4x^2+1)` has no Real factors.