$\displaystyle (a^2n-a^n-6)/(a^n+8)$
Divide.
Hello, A Beautiful Mind!
Is there a second typo?
The problem would make more sense . . .
Divide: .$\displaystyle \frac{a^{2n}-a^n-6}{a^{{\color{red}3}n}+8}$
Factor: .$\displaystyle \frac{(a^n+2)(a^n-3)}{(a^n+2)(a^{2n}-2a^n + 4)} $
Reduce: .$\displaystyle \frac{a^n - 3}{a^{2n} - 2a^n + 4} $
Yes, the first typo was the exponent "2n" . . .
You have: .$\displaystyle (a^{{\color{red}2n}} - a^n - 6) \div (a^n + 8)$
Long division:
. . $\displaystyle \begin{array}{cccccccc}
&&&& a^n & - & 9 \\
& & -- & -- & -- & --& -- \\
a^n + 8 & ) & a^{2n} & - & a^n & - & 6 \\
& & a^{2n} & + & 8a^n \\
& & -- & -- & -- \\
& & & - & 9a^n & - & 6 \\
& & & - & 9a^n & - & 72 \\
& & & -- & -- & -- & -- \\
& & & & & & 66 \end{array}$
Answer: .$\displaystyle a^n - 9 + \frac{66}{a^n+8}$