Question

How do you write `4x^2+2x-12` in factored form?

Answer 1

`4x^2+2x-12`
`color(white)("XXXXXX")=color(green)(2)color(red)((2x+3))color(blue)((x-2))`

Explanation:

Extract the obvious constant factor of `2` to simplify

`color(green)(2)color(orange)((2x^2+x-6))`

Factoring
by looking for integer factors `a` and `b` of `2`
and integer factors `c` and `d` of `(-6)`
such that
`(ad+bc)x = 1x`

The only integer factors of `2` are `1xx2`

Integer factors of `(-6)` are `{(1xx-6),(2xx-3)(-1xx6),(-3xx2)}`

Checking the four possible combinations, we find:
`color(orange)((2x^2+x-6))=color(red)((2x+3))color(blue)((x-2))`

So `4x^2+2x-12`
`color(white)("XXX")=color(green)(2)color(orange)((2x^2+x-6))`
`color(white)("XXX")=color(green)(2)color(red)((2x+3))color(blue)((x-2))`