Question

How do you factor `4x^2+4x-3`?

Answer 1

Playing with the factors for `4` and `3` gives:

`(2x - 1)(2x + 3)`

Answer 2

The only front factors of `4` that look convincing are `2` and `1`. We assume a factorization of the form

`(2x pm ?)(2x pm ?)`

The `3` term is negative, so we have

`(2x + ?)(2x - ?)`

The factors of `-3` are only `pm3` and `∓1`. So, we have to decide whether it's `+3` and `-1`, or `-3` and `+1`.

Since the `4x` term is positive, the terms in "`?`" are likely `+3` and `-1` rather than `-3` and `+1`, allowing the positive term to dominate the sum of the outer and inner factors.

`=> color(blue)((2x + 3)(2x - 1))`

---

Verify that this is correct by multiplying this out again. We could have been wrong and we could have wrongly chosen the form

`(4x pm ?)(x pm ?)`...

This would not have been correct:

`(4x - 3)(x + 1)`

The middle term would not be `4x`, even though the first and last terms would have been correct.