The Hardest Factoring  Monomial Factoring 

 You must end with as many terms in the parenthises as you started with in the original unfactored expression.
 1st. Find the common factors.
 2nd. Copy the common factors "out front"  outside the parenthises.
 3rd. Copy what's left as terms in the parenthesis.
 4th. Check by multiplying.
Factoring Trinomials  Written Instructions 

Factoring Trinomials (animation)
 With long notes:
 Factor: x^{2}  4x  21
 1st: Arrange the trinomial in descending order.
 x^{2}  4x  21

 2nd: Set up the answer.
 x^{2}  4x  21
 (x )(x )

 3rd: List all factor pairs of the last (constant) term.
 21
 1 , 21
 3 , 7

 4th: Add or subtract to get the middle term.
 21
 1 , 21
 + 3 ,  7 will produce the  4.

 Since:
 (+)() = ()
 ()(+) = () produces a negative and
 (+)(+) = (+)
 ()() = (+) produces a positive,
 The  21 is created by multiplying one negative & one positive.

 This rule always works.
 If the constant term is + the factors have the same sign, so ADD the factors to find the middle term.
 If the constant term is  the factors have different signs, so SUBTRACT the factors to find the middle term.

 In short, Check the sign of the middle (linear) term
 If + , ADD to find the middle term.
 If  , SUBTRACT to find the middle term.

 5th: The "larger" number always takes this (linear term) sign.
 x^{2}  4x  21 The  4x is negative, so the 7 gets a negative sign.
 (x + 3)(x  7)

 6th: Multiply to check answer.

 With short notes:
 Factor: x^{2}  4x  21
 1st: Arrange the trinomial in descending order.
 x^{2}  4x  21
 2nd: Set up the answer.
 x^{2}  4x  21
 (x )(x )
 3rd: List all factor pairs of the last (constant) term.
 21
 1 , 21
 3 , 7

 4th: Add or subtract to get the middle term.
 If + , ADD to find the middle term.
 If  , SUBTRACT to find the middle term.
 21
 1 , 21
 + 3 ,  7 will produce the  4.

 5th: The "larger" number always takes this (linear term) sign.
 x^{2}  4x  21 The  4x is negative, so the 7 gets a negative sign.
 (x + 3)(x  7)
 6th: Multiply to check answer.
Factoring Trinomials when The Leading Coefficient Is Not One 
Factoring Trinomials when The Leading Coefficient Is Not One (animation)
 1st: Set up the answer.
 2nd: List all factor pairs of the last term.
 3rd: List all factor pairs of the first term.
 4th: List all factor pairs of the product of the first and the last terms.
 5th: Add or subtract to get the middle term.
 If + , add and make both the same sign.
 If  , subtract and make the signs different.
 6th: The "larger" number always takes this (linear term) sign.
 7th: Match a pair of factors of the first with a pair of factors of the last to make the required factors of the product .
 8th: Place factor by matching OUTERS with OUTERS and INNERS with INNERS.
 9th: Verify by multiplying to obtain the product.
