1. **State the problem:** Subtract the two rational expressions: $$\frac{6}{x^2 + 4x - 32} - \frac{x - 5}{x - 4}$$ 2. **Factor the denominator of the first fraction:** $$x^2 + 4x - 32 = (x + 8)(x - 4)$$ 3. **Rewrite the expression with factored denominators:** $$\frac{6}{(x + 8)(x - 4)} - \frac{x - 5}{x - 4}$$ 4. **Find a common denominator:** The least common denominator (LCD) is $(x + 8)(x - 4)$. 5. **Rewrite the second fraction with the LCD:** $$\frac{x - 5}{x - 4} = \frac{(x - 5)(x + 8)}{(x - 4)(x + 8)}$$ 6. **Subtract the numerators over the common denominator:** $$\frac{6 - (x - 5)(x + 8)}{(x + 8)(x - 4)}$$ 7. **Expand the numerator:** $$(x - 5)(x + 8) = x^2 + 8x - 5x - 40 = x^2 + 3x - 40$$ So numerator becomes: $$6 - (x^2 + 3x - 40) = 6 - x^2 - 3x + 40 = -x^2 - 3x + 46$$ 8. **Final expression:** $$\frac{-x^2 - 3x + 46}{(x + 8)(x - 4)}$$ 9. **Check if numerator can be factored:** The quadratic $-x^2 - 3x + 46$ does not factor nicely with integer roots. **Answer:** $$\boxed{\frac{-x^2 - 3x + 46}{(x + 8)(x - 4)}}$$