Question

How do you factor `15 + m - 6m^2`?

Answer 1

You have to find the roots of this quadratic expression and transform them into factors, as follows:

Using Bhaskara to find the roots:

`(-1+-sqrt(1-4(-6)(15)))/(-12)`
`(-1+-19)/-12`
`m=5/3` and `m=-3/2`

Keeping the equatlity, we can rewrite these answers (roots) as factors:

`3m-5=0` and `-2m-3=0`

So, your equation factored is:

`(3m-5)(-2m-3)`

Answer 2

Multiply `15xx-6m^2 = -90m^2`

We need two temrs that multiply to give us `-90m^2` and that add to give us `m = 1m`.
A little thought convinces us that to get a negative when we multiply, we need one positive and one negative number.
To get a positive when we add, the number with the larger absolute value must be the positive number.

Try them:

`-1mxx90m` does not add up to `1m`

`-2mxx45m` does not add up to `1m`

`-3mxx30m` does not add up to `1m`

`4` is not a factor of `90`

`-5mxx18m` does not add up to `1m`

`-6mxx45m` does not add up to `1m`

`7` is not a factor of `90`
`8` is not a factor of `90`

`-9mxx10m` STOP! this does add up to `1m`

Split the middle term (the `1m`) using the ywo we just found. (Either order will work.)
Then factor by grouping:

`15+m-6m^2 = 15 -9m + 10m -6m^2`

`color(white)"sssssssssssssss"` `= (15 -9m) + (10m -6m^2)`

`color(white)"sssssssssssssss"` `= 3 (5 -3m) + 2m(5 -3m)`

`color(white)"sssssssssssssss"` `= (3 + 2m)(5 -3m)`

Check the answer by multiplying.

Answer 3

I use the new AC Method (Google, Yahoo Search) to factor trinomials.

`f(m) = -6m^2 + m + 15 = a(m -p)(m - q)`

Converted function:`f'(m) = m^2 + m - 90 = (m - p')(m - q')`

Find p' and q' by composing factor pairs of a.c = -90. Proceed: ...(-6, 15)(-9, 10). This last sum is -9 + 10 = 1 = b. Then (p') = -9' and (q') = 10.
Back to original function: `p = (p')/a = -9/-6 = 3/2` and `q = (q')/a = 10/-6 = -5/3.`
Factored form: `f(m) = -(m + 3/2)(m - 5/3) = -(2m + 3)(3m - 5)`.

Chec by developing:
`f(m) = -(6m^2 - 10m + 9m - 15) = -6m^2 + m + 15`. OK