Question

How do you factor `6x^2-13x-28`?

Answer 1

`(3x+4)(2x-7)`

Explanation:

For your `x-`terms you could have: `2x` & `3x` or `6x` & `x`

For your constants, you need two numbers that have a product of `-28`. This means one must be positive and one must be negative. Since the `x`-term = `-13x` is negative, the larger constant needs to be negative.
Here are your choices: `1` & `-28`, `2` & `-14`, `4` & `-7`

Since the `x`-term = `-13x` is fairly small, select the constants that are close to each other: `4` & `-7`

Form possible factors:
`(6x +4)(2x - 7)` gives the `x`-term `= -42x + 8x = -34x` too big

`(3x+4)(2x - 7)` gives the `x`-term `= -21x + 8x = -13x` correct

Therefore the factors are: `(3x+4)(2x-7)`

Answer 2

`6x^2-13x-28 = (3x+4)(2x-7)`

Explanation:

Given:

`6x^2-13x-28`

Use an AC method:

Find a pair of factors of `AC = 6*28 = 168` which differ by `B=13`.

[[ Note that we look for a given difference rather than sum, since the sign on the constant term is negative ]]

Since the difference we want is odd, one of these factors must contain all of the powers of `2` in `168` - that is be a multiple of `8`.

We find that the pair `21, 8` works.

Use this pair to split the middle term and factor by grouping:

`6x^2-13x-28 = 6x^2-21x+8x-28`

`color(white)(6x^2-13x-28) = (6x^2-21x)+(8x-28)`

`color(white)(6x^2-13x-28) = 3x(2x-7)+4(2x-7)`

`color(white)(6x^2-13x-28) = (3x+4)(2x-7)`