Why is 12 (m+n) equivalent to 12m + 12n

Because 12(m=n) means that you are multiplying 12 times m and 12 times n. So if there is a number outside of parenthesis it gets multiplied by all the nu,beds in the parenthesis.
12(1+2)
36 either way. that is proof. if you want more proof use any number in replace of n and m

Because of the distributive property. You multiply the outside number (12) by the variables inside the parentheses (m and n). Then you add them together, so you get 12m+12n. 
You can check it by plugging in numbers for m and n, then solving it both ways. You will get the same answer.