In basketball there are three ways to score. 1 point for a free throw, 2 points for a short field goal, and 3 points for a long field goal. during the first quarter of a regular season basketball game, the dallas mavericks scored 36 points but did not make any free throws. what is the other extreme case in table 1.1 the entry where sfg=0 and how can you systemically record the other entries in table 1.1

It says Dallas didn't make any free throws. It means there are short field goals (SFG) and long field goals (LFG) only.
Let's assume number of SFG = x
and number of LFG = y

Given that 2 points for a short field goal, and 3 points for a long field goal.
Total Score = 2x+3y
Also, Total score = 36 (given in the question)
So we have the equation

where x and y are integers.

when SFG=0, then x=0 and so 3y=36.
y=12 i.e. LFG = 12
So the extreme case is (0,12) i.e. SFG=0 and LFG=12.

Other cases would be integral vales of (x,y) which satisfy the equation 2x+3y=36.
So (3,10) , (6,8) , (9,6) , (12,4) , (15,2) , and (18,0) are other cases.

Hence final answers are (x,y) = (0,12) , (3,10) , (6,8) , (9,6) , (12,4) , (15,2) , and (18,0)

4should be the last one n=4