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It's not clear whether the problem is to:

(1) divide \(x^2-4\) by \(\dfrac{x-8}{x-2}\), or

(2) divide \(\dfrac{x^2-4}{x-8}\) by \(x-2\).

But with only two possible cases, that's not much work to do.

\(\drac{x^2-4}{\frac{x-8}{x-2}}=\dfrac{(x^2-4)(x-2)}{x-8}=\dfrac{x^3-2x^2-4x+8}{x-8}\)

Synthetic division yields a quotient and remainder of \(x^2+6x+44+\dfrac{360}{x-8}\) (see attachment for how the algorithm was carried out).

In the other case,

\(\dfrac{\frac{x^2-4}{x-2}}{x-8}=\dfrac{\frac{(x-2)(x+2)}{x-2}}{x-8}=\dfrac{x+2}{x-8}\)

provided that \(x eq2\). Synthetic division can be used again here, but this is quite easy to deal with by just doing some easy manipulation of the numerator:

\(\dfrac{x+2}{x-8}=\dfrac{x-8+10}{x-8}=\dfrac{x-8}{x-8}+\dfrac{10}{x-8}=1+\dfrac{10}{x-8}\)

(also provided that \(x eq 8\))