Here is the initial expression and the steps I've made so far, but from the final line I can't go on. Have I made a mistake somewhere?
$$\frac{a+2}{a^3-8}-\frac{1}{a^2-a-2}=\frac{a+2}{(a-2)(a^2+2a+4)}-\frac{1}{(a-2)(a+1)}=\frac{(a+2)(a+1)-(a-2)(a^2+2a+4)}{(a-2)(a^2+2a+4)(a+1)}$$
Corrected to match the corrected question: Your last numerator should be just $(a+2)(a+1)-(a^2+2a+4)$, without the factor of $(a-2)$ in the second term, since you already have a factor of $a-2$ in the second denominator. Then the final fraction simplifies significantly.