Question

How do you factor `12x^2 - 5x - 3`?

Answer 1

We start by hoping for integer coefficient terms for our factors
and note
factors of `(6) = {(1,12), (2,6), (3,4)}`
factors of `(3) = {(1,3)}`

We are looking for
`(ax+-b)(cx+-d)`
where `(a,c)` are factors of `6`
and `(b,d)` are factors of `3`
such that `ad-bc = -5`

With the limited pairs of factors which we have predetermined (and some juggling of plus/minus signs)
we get
`12x^2-5x-3 = (3x+1)(4x-3)`

Answer 2

You can find its roots (using Bhaskara, in this resolution) and then turn them into factors, by equaling each to zero, as follows.

`(5+-sqrt(25-4(12)(-3)))/24`
`(5+-13)/24`
`x_1=3/4`, which can be rewritten as the factor `(4x-3)=0`
`x_2=-1/3`, which can be rewritten as the factor `(3x+1)=0`

So

`12x^2-5x-3=(4x-3)(3x+1)`