Question

How do you solve `3x^2+5x+2=0` by factoring?

Answer 1

Assuming `3x^2+5x+2` can be factored into
`(ax+b)(cx+d)` with integer values for `a,b,c,d`
and noting that all of `a,b,c,d` will be non-negative (from the form of the original quadratic)
we only have the possibilities
`{a,c} = {1,3}` and
`{b,d} = {1,2}`

There are only 2 combinations to attempt:
`(1x+1)(3x+2)`
and
`(1x+2)(3x+1)`

revealing that `3x^2+5x+2 =0` can be factored as
`(x+1)(3x+2)=0`

So either
`x+1 = 0` which implies `x = -1`
or
`3x+2 = 0` which implies `x = -2/3`