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?
