1. How many different ways can you factor a trinomial?
This is intended as a tutorial for teachers to watch. Watch as a slideshow with animations, and it is much easier to follow.
I recommend teaching your students 2 ways maximum. Otherwise I think it would be more confusing than helpful.
Once you decide, alter the PowerPoint to meet your needs.  Happy Teaching!

2. The X (Magic-X, Big-X, Amazing-X, X-marks the spot) Whatever you want to call it… This will, by itself, factor a trinomial when A=1… <Then you just add steps when A≠1>
Students should understand standard form Ax2+Bx+C
A•C
Attic& ceiling are up
These two numbers must multiply to make A•C and add to make B
CHECK SIGNS!
x2-3x-10
a•c
Both ( ) will start with x,
Then use the two side numbers to fill in. 10 ● +2 -5 + -3 b
( ) ( )
x+2
x-5 B
Basement is down

3. Way 1: The Advanced X This seems to be the method of choice for many of the high school teachers because then you only have to teach method and it covers all cases! I am including several examples of this method.
Bring down the “A” term to both sides, but without the ”2”
y2-3y-4
a•c
Treat each as a fraction and reduce. -4 1y 1y 1 -4 -3 b
Turn the “fractions” into factors
Here is your answer!
(y-4)
(y+1)

4. Way 1: The Advanced X 15y2-16y+1 (y-1) (15y-1) 15y 15y
Bring down the “A” term to both sides, but without the ”2”
15y2-16y+1
a•c
Treat each as a fraction and reduce. 15
15y
15y -1
-15
-16 b
Turn the “fractions” into factors
Here is your answer!
(y-1)
(15y-1)

5. Way 1: The Advanced X 15y2-40y+25 5(3y2-8y+5) 5 (3y-5) (y-1) 3y 3y
Factor first- if you can
15y2-40y+25
Bring down the “A” term to both sides, but without the ”2”
5(3y2-8y+5)
a•c
Treat each as a fraction and reduce. 15 3y 3y -3 -5 -8 b
Turn the “fractions” into factors
Here is your answer! 5
(3y-5)
(y-1)
Don’t forget the factor you pulled out!

6. Move factors over to box
Way 2: The X-Box
Move factors over to box
15y2-16y+1
15y -1 1
15y2
a•c
Factor across y 15
-15y
-15 y
-1y -1 y
-16 -1
Factor across b
Factor up
Factor up
( ) ( ) y -1
15y -1

7. Way 3: X, then ladder (elephant ears)
Down on the left
Factor out
Up on the right
15y2-16y+1
15y2 +1
15y
a•c 15
This is what’s left
y -1
-15y
-1y
-15 y -1 y
Factor out
-16 b -1
This is what’s left
Move the factors over to the ladders
y -1
( ) ( ) y -1
15y -1

8. Way 4: X, then group 15y2-16y+1 15y2 +1 -15y -1y 15y (y-1) + -1 (y-1)
a•c 15
-15y
-1y
-15 y -1 y
Move factors, then group them.
-16 b
Pull out common factors from each group
Use “reverse” distribution
15y
(y-1) + -1
(y-1)
(y-1)
(15y-1)

9. Way 5: X, then divide 15y2-16y+1 (15y-15) (15y-1) 15 (y-1) (15y-1)
Group factors- move them over.
If either group can be divided, do so.
15y2-16y+1
a•c
(15y-15)
(15y-1) 15 -1
-15 15
-16
This is what’s left. b
(y-1) (15y-1)

10. Way 5: X, then divide- example 2 (oops, I should have factored first, but didn’t!)
Group factors- move them over.
If either group can be divided, do so.
3x2+9x-12
If both groups can be divided by the same thing, it goes in front.
a•c
(3x-3)
(3x+12) 36 12 -3 3 3 9 b 3
(x-1) (x+4)
This is what’s left.