The length of the segment between the points $(2a, a-4)$ and $(4, -1)$ is $2\sqrt{10}$ units. What is the product of all possible values for

Your questions seem to  be  truncating, but I can answer this one.....

Using the distance  formula....

sqrt  [ (2a - 4)^2   + (a - 4 - (-1) )^2  ] =  2sqrt (10)        square both sides

(2a - 4)^2  +  ( a - 3)^2  =  40    simplify

4a^2 - 16a + 16  + a^2 - 6a + 9 =  40

5a^2 -22a  -15  =  0       factor

(5a  - 3) ( a  + 5)  =  0

So

5a - 3  =  0           a +  5   =0

5a   = 3                 a  = -5

a = 3/5

Product of all possible values of a =  (3/5) (-5)  =   -3