What is the solution of -8/2y-8=5/y+4 - 7y+8/y^2-16? y = –4 y = –2 y = 4 y = 6

Please clarify whether you meant (-8/2)y or   -8 / (2y - 8).

It's always a good idea to use parentheses for clarity; it reduces or removes the need for guessing.

Here is my interpretation of your equation:

-8/2y-8=5/y+4 - 7y+8/y^2-16  becomes

-8 / (2y-8) = 5 / (y+4) - 7y + 8 / (y^2-16)   (note use of parentheses)

Factoring all of the denominators results in:

-8 / [2(y-4)] = 5 / (y+4) - 7y  + 8 / [(y-4)(y+4)]

The LCD is 2(y-4)(y+4).  You must multiply each and every term in the above equation by this LCD to clear fractions:

-8(y+4) = 5(2)(y-2) - 7(y+4)(y-4) + 8/2

Then -8y - 32 = 10y -20 -7y^2 + 4

I believe the "-7y^2" term is incorrect.

In retrospect, you could simply substitute each of the given answers (y-values), one at a time, to determine which, if any, makes the equation true:

-8 / (2y-8) = 5 / (y+4) - 7y + 8 / (y^2-16)   (note use of parentheses)

Let's test y = 4.  Is this a solution?  Subst. 4 for y in the above equation, we get:

-8 / (8-8) and several more terms.  No, 4 is not a solution, because 
-8 / (0) is undefined.  Eliminate y = 4 as a solution.  Try y = -4 and the other given possible solutions.  Do any of them make this equation true?