Fully factor x² + 8x + 12

When looking at this polynomial, we are looking for two numbers that multiply to a positive 12 and add to a positive 8

List all the factors of 12: (1,12) (2,6) (3,4)

Out of all the factors, 2 and 6 add up to 8

Therefore, the factored binomials are (x + 2) and (x + 6) 

helpful hint: You can check your work by multiplying the two back together and you should get x² + 8x + 12

Fully factor 4x² – 19x – 5

We cannot neglect the 4 coefficient, therefore we need to use the AC method

We multiply the value of A (4) by the value of C (-5) to get a resulting value

4 ⋅ -5 = -20

Then we find a factor of -20 that add to a -19

Factors of 20: (1,20) (2,10) (4,5)

Looking at the factors of 20, we can see that if we added -20 and a positive 1, we would get -19

Break apart the -19x into component parts

4x² – 20x + 1x – 5

The value of our expression did not change, just took on a different format

We can then do what is called factor by grouping. Group the expression together in pairs that can be factored

(4x² – 20x) + (x – 5)

Then factor out the monomials accordingly

4x (x – 5) + (x – 5)

We want the two binomials that are left to match. They do in this case so we are able to move the 4x and 1 into their own binomial

4x (x – 5) + 1(x – 5) 

(4x + 1) (x – 5) 

The two binomials that 4x² – 19x – 5 factor into are (4x + 1) (x – 5)