Question

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

Answer 1

`3x^2-5x-2=color(blue)((1x-2)(3x+1))`

Explanation:

Looking for `{a,b,c,d}` for a factoring `(ax+c)(bx+d)=3x^2-5x-2`

We see that
`color(white)("XXX")ab=3`
`color(white)("XXX")cd=-2`
and
`color(white)("XXX")ad+cb=-5`

In the hopes of finding integer values for `{a,b,c,d}`
we note that the factors of `3` are `{1,3}`
so if `a, b` are integers `{a,b}={1,3}`
and
we also note the the factors of `-2` are `{(-1,2),(1,-2)}`

We can test these possible combinations looking for `ad+cb=5`

#{: (underline(a),underline(b),underline(c),underline(d),,underline(ad+cb)), (1,3,-1,2,,-1), (1,3,2,-1,,+5), (1,3,1,-2,,+1), (color(red)(1),color(red)(3),color(red)(-2),color(red)(1),,color(red)(-5)) :}#

We have found values that match our requirements:
`color(white)("XXX")< a,b,c,d > = < 1,3,-2,1 >`
So
`color(white)("XXX")(ax+c)(bx+d)=(1x-2)(3x+1)`