Use long division or synthetic division to divide the given polynomial.

To divide the polynomial \( x^3 - 17x + 32 \) by \( x + 5 \) using long division, follow these steps:

1. Set up the long division:

   • Divide \( x^3 \) by \( x \), which gives \( x^2 \).
   • Multiply \( x^2 \) by \( x + 5 \) to get \( x^3 + 5x^2 \).
   • Subtract this from the original polynomial:

   \[ (x^3 - 17x + 32) - (x^3 + 5x^2) = -5x^2 - 17x + 32 \]

2. Next, bring down the terms:

   • Divide \( -5x^2 \) by \( x \), which gives \( -5x \).
   • Multiply \( -5x \) by \( x + 5 \) to get \( -5x^2 - 25x \).
   • Subtract this from the current polynomial:

   \[ (-5x^2 - 17x + 32) - (-5x^2 - 25x) = 8x + 32 \]

3. Next, continue with the division:

   • Divide \( 8x \) by \( x \) to get \( 8 \).
   • Multiply \( 8 \) by \( x + 5 \) to get \( 8x + 40 \).
   • Subtract this from the current polynomial:

   \[ (8x + 32) - (8x + 40) = -8 \]

4. Now, we have our results:

   The quotient is \( x^2 - 5x + 8 \) and the remainder is \(-8\).

This means:

\[ \frac{x^3 - 17x + 32}{x + 5} = x^2 - 5x + 8 - \frac{8}{x + 5} \]

Thus, the correct answer is B: \( x^2 - 5x + 8 - \frac{8}{x + 5} \).