[tex]1 \div 3(12 + x) = - 8(6 - x)[/tex]

Buckle up, this one was a long one. 

I set up 1 / 3(12 + x) as a fraction and set it equal to - 8(6 - x). 

Get rid of the parentheses by distributing the integers on the outside. You get 1 / 36 + 3x = - 48 + 8x. 

To get rid of the fraction, you multiply both sides by the (36 + 3x) polynomial to get 1 = (- 48 + 8x)(36 + 3x). 

Now we do the FOIL method to get 1 = 24x^2 + 288x - 144x - 1728. Combine like terms to get 24x^2 + 144x - 1728 = 1. 

Simply again by dividing each term the GCF of 24 to get 24(x^2 + 6x - 72) = 1. 

Factor the inside of the parentheses to get 24(x + 12)(x - 6) = 1. Now divide both sides by 24 to isolate the x variables to get (x + 12)(x - 6) = 1/24.

Set each polynomial equal to 1/24 to get you possible answers for what x can be. 

x + 12 = 1/24. Subtract 12 from both sides; you get x = - 12 + 1/24 to get x = - 11 23/24. 

x - 6 = 1/24. Add 6 to both sides; you get x = 6 + 1/24 to get x = 6 1/24