Step-by-step explanation:
Consider this polynomial,
![8 {x}^{2} + 2x - 3](https://tex.z-dn.net/?f=8%20%7Bx%7D%5E%7B2%7D%20%20%2B%202x%20-%203)
First, I do the AC memethod. To find my two numbers,
That is to find two numbers that multiplied will equal my leading coeffeicent, A times my constant, C
And that will also add to 2, my middle term.
So using that
![8 \times - 3 = - 24](https://tex.z-dn.net/?f=8%20%5Ctimes%20%20-%203%20%3D%20%20-%2024)
So let find factors of-24:
Factors can be positve or negative
The factors of 24 are : 1,2,3,4,6 8,12,24.
So which one of these add up to 2.
![6 + ( - 4) = 2](https://tex.z-dn.net/?f=6%20%2B%20%28%20-%204%29%20%3D%202)
and
![6 \times - 4 = - 24](https://tex.z-dn.net/?f=6%20%5Ctimes%20%20-%204%20%3D%20%20-%2024)
So six and negative four is our numbers.
Now, let set up our binomials
Step 1: Rewrite the orginal equation by using 6x and -4x instead of 2x.
Disclaimer: The placement of these numbers doesn't matter.
So we have
![8 {x}^{2} + 6x - 4x - 3](https://tex.z-dn.net/?f=8%20%7Bx%7D%5E%7B2%7D%20%20%2B%206x%20-%204x%20-%203)
Step 2: Group the first two terms and last two terms
![(8 {x}^{2} + 6x) + ( - 4x - 3)](https://tex.z-dn.net/?f=%288%20%7Bx%7D%5E%7B2%7D%20%20%2B%206x%29%20%2B%20%28%20-%204x%20-%203%29)
Factors the first group by finding the GCF.
x is the greatest common variable, and 2 is the gcf so we have
![2x(4x + 3) + ( - 4x - 3)](https://tex.z-dn.net/?f=2x%284x%20%2B%203%29%20%2B%20%28%20-%204x%20-%203%29)
Next, factor the next group by -1.
![2x(4x + 3) - 1(4x + 3)](https://tex.z-dn.net/?f=2x%284x%20%2B%203%29%20-%201%284x%20%20%2B%203%29)
Combine the outside factors.
![(2x - 1)(4x + 3)](https://tex.z-dn.net/?f=%282x%20-%201%29%284x%20%2B%203%29)
So those are the factors.