Question

How do you factor `3y^2+7y-6`?

Answer 1

`3y^2+7y-6 = (3y-2)(y+3)`

Explanation:

You could use the quadratic formula for this
but (hoping for integer coefficient factors):

Since `(ay+b)(cy+d) = (ac)y^2 + (ad+bc)y +bd`

and for the given example
`color(white)("XXXX")`(ac) = 3`and`(bd) = (-6)#

we want to find `(a,c)`, factors of 3
`color(white)("XXXX")`only obvious factors are [(1,3), (3,1)]
and factors, `(b,d)`,of (-6)
`color(white)("XXXX")`[(1,-6), (-1,6), (2,-3), (-2,3), (6, -1), (-6,1), (3,-2), (-3,2)]
which satisfy
`color(white)("XXXX")`(ad+bc) = 7#

with a small amount of effort we find
`color(white)("XXXX")` `(a,c) = (3,1)` and `(b,d) = (-2,3)`