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It's not clear whether the problem is to:

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

(2) divide [tex]\dfrac{x^2-4}{x-8}[/tex] by [tex]x-2[/tex].

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

[tex]\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}[/tex]

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

In the other case,

[tex]\dfrac{\frac{x^2-4}{x-2}}{x-8}=\dfrac{\frac{(x-2)(x+2)}{x-2}}{x-8}=\dfrac{x+2}{x-8}[/tex]

provided that [tex]x\neq2[/tex]. Synthetic division can be used again here, but this is quite easy to deal with by just doing some easy manipulation of the numerator:

[tex]\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}[/tex]

(also provided that [tex]x\neq 8[/tex])