How do you factor by grouping `30x^2 + 76x + 48`?

`30x^2+76x+48`

`= 2(15x^2+38x+24)`

`= 2(15x^2+20x+18x+24)`

`= 2((15x^2+20x) + (18x+24))`

`= 2(5x(3x+4)+6(3x+4))`

`= 2(5x+6)(3x+4)`

How did I find a splitting of the middle `38x` term into `20x+18x` that would work?

The coefficients of `15x^2+38x+24` are:

`A=15`, `B=38`, `C=24`

We need to find a factorization of `AC=15*24=360` into a pair of factors whose sum is `B=38`.

`15+24 = 39` is quite close to the `38` we want, so the pair of numbers we're looking for will be similarly close to one another. In fact, notice that `361 = 19^2` and `19+19 = 38` so
`(19-1)(19+1) = 19^2-1^2 = 361 - 1 = 360` as required.

`f(x) = 2(15x^2 + 38x + 24).`
If you like to avoid guessing, or avoid the lengthy factoring by grouping, use the systematic New AC Method.
`f(x) = 2a(x - p)(x - q)`
Convert f(x) to f'(x) = x^2 + 38x + 360. = (x - p')(x - q'). Compose factor pairs of a.c = 360. a and c have same sign. Proceed:...(15, 24)(18, 20). OK. This sum is 18 + 20 = 38 = b. Then, p' = 18 and q' = 20
Then `p = (p')/a = 18/15 = 6/5, and q = 20/15 = 4/3.`

Factored form:`f(x) = 30(x + 4/3)(x + 6/5) = 2(3x + 4)(5x + 6)`