1)  Look for a greatest common factor first.  Put it outside parentheses.

18r²s³ + 12r⁴s

6r²s(3s² + 2r²) 

-24pr⁵ - 56p³

-8p(3r⁵ + 7p²)  

2)  If there are 2 terms, look for a difference of 2 squares or a sum/difference of 2 cubes:

16k² - 1        difference of 2 squares, plus minus pattern

(   +   )(   -   )

(4k+1)(4k-1)

27-y³                difference of 2 cubes, binomial has a minus, trinomial has 2 plusses

(3 - y)(9 + 3y + y²)

a³ + b³             sum of 2 cubes, binomial has a plus, trinomial has minus and plus

(a + b)(a² - ab + b²)

3)  If there are 3 terms, try two sets of parentheses.  Decide on the signs:  ++, - - , or + -.  Will the middle term be a sum(if the signs are the same) or a difference (if the signs are different)?   Then put in the variables, numbers, and check it with FOIL.

8x² + 14x + 3                                 x² - 12x + 36                         x² - 2x - 15

(    +    )(    +    )                           (   -    )(   -   )                        (   +   )(   -   )

(4x + 1)(2x + 3)                            (x - 6)(x - 6)                         (x + 3)(x - 5)

                                                        (x - 6)²

Check your factoring by multiplying out the factors using FOIL.  If you factored correctly, you should end up with what you started with.

4)  If there are 4 or more terms, try grouping:

ab - 2 - 2b + a                                             u(v - 1) - 2(1 -v)        [factor  out a (-1) from (1 - v)]

ab + a - 2b - 2                                             u(v - 1) +2(v - 1)       [(-2)(-1)= +2]

a(b + 1) -2(b + 1)                                        (v - 1)(u + 2)

(b + 1)(a - 2)

5)  Check to make sure that all plusses and minuses are inside parentheses.  Check to make sure that each factor is prime.  If a factor occurs more than once, write it once with an exponent.