understanding plus/minus in residues

Question

I am trying to understand how they get the value for the residue given that its plus/minus value for the zeros. If you were to simply plug in positive i/root2 and negative i/root2 thereafter you still don't get the desired result. I have been struggling on this concept for long, could someone please explain to me

Answer 1

$\frac{z}{8z^3 +10z} = \frac{1}{8 z^2 +10}$ and you can see that the sign doesn't affect the answer (since we are squaring $z$). If you plug in you do indeed get $1/6$

Answer 2

The given function is odd, hence if $w$ is a pole $-w$ is also a pole and the residues at such points are the same. $\pm$ is just a shorthand notation for writing two results in a single line, namely $$ \text{Res}(f(z),z=+w) = \frac{1}{6} = \text{Res}(f(z),z=-w).$$