Verified answer

x^2 / (x^2 - 64) = x^2 / ((x+8)(x-8))

(x+4)/(x+8) = ((x+8)-4)/(x+8) = 1 - 4/(x+8)

Since x^2 - 64 = (x + 8)(x - 8), you multiply top and bottom of the second expression by (x - 8) and then subtract the tops:

x^2 - (x + 4)(x - 8)

x^2 - (x^2 - 4x - 32)

= x^2 - x^2 + 4x + 32

= 4x + 32

= 4(x + 8) so it's that over (x + 8)(x - 8)

and the (x + 8)'s cancel

so just 4 over (x - 8)