How do you factor `3x^2-22x-16`?

First multiply your `a and b` terms to get `-48`
Then find two numbers that multiply to that and add up to `-22`
Once you do that rewrite you `b` term as those two numbers like so :
`3x^2 +2x - 24x -16`
Then put parenthesis around your first and last two terms so you can factor them:
`(3x^2 +2x) (- 24x -16)`
Then factor normally:
`x(3x+2) -8(3x+2)`
your two parenthesis should be the same and if they are cross out one of them and make a new parenthesis with the two numbers you factored out:
`(3x+2)(x-8)`

We begin with `3x^2-22x-16` Automatically we can set up this..
`(3x+?)(x+?)`Since `3x^2` could only factor into `3x` and `x`
Now to fill in the ?'s
The question marks must multiply to equal 16 and add up to -22.
You can always use the guess and check method until you come across the correct answer or set up some equations..
`3x+y=22`
`xy=-16`
Then to solve this system of equations you can use subsitution...

These are not always easy to spot. Sometimes you really have to 'play' with them to find the correct combination.

First lets look at the constant of `-16`

As this is negative it means that it is the product of a negative and a positive value.

What are all the whole number factors of 16?

`1xx16; 2xx8; 4xx4`

3 is a prime number so we can only have `1xx3` giving:

`(x+-?)(3x+-?)`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
I am deliberately 'going round the houses' to make a point!

`color(brown)("Try 1 and 16 for "-22x)`

`(x+-?)(3x+-?) -> (x+-1)(3x+-16)`

`+-16+-3!=-22: color(white)("dd") +-(3xx16)+-1=-22 color(red)(larr"Fail")`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(brown)("Try 4 and 4 for "-22x)`

`(x+-?)(3x+-?) -> (x+-4)(3x+-4)`

`+-(3xx4)+-4!=-22 color(red)(larr" Fail")`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(brown)("Try 2 and 8 for "-22x)`

`(x+-?)(3x+-?) -> (x+-2)(3x+-8)`

`(3xx2)-8!=-22 color(red)(larr" Fail")`

`(3xx(-8))+2 =-22 color(green)(larr" WORKS!")`

`(x-8)(3x+2)`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Check")`

`+x(3x+2) = 3x^2+2x`
`-8(3x+2)=ul(color(white)(ccd.) -24x-16)`
`color(white)("dddddddddddd")3x^2-22x-16 larr" As required"`