1. **State the problem:** Simplify the expression $$\frac{y + 12}{x - 8} + \frac{y(x - 8)}{x^{2}y - 8xy}$$ and determine which given option it is equivalent to. 2. **Analyze the denominators:** - The first denominator is $$x - 8$$. - The second denominator is $$x^{2}y - 8xy$$. 3. **Factor the second denominator:** $$x^{2}y - 8xy = xy(x - 8)$$. 4. **Rewrite the expression with factored denominator:** $$\frac{y + 12}{x - 8} + \frac{y(x - 8)}{xy(x - 8)}$$. 5. **Simplify the second fraction:** Cancel $$x - 8$$ in numerator and denominator: $$\frac{y(x - 8)}{xy(x - 8)} = \frac{y}{xy} = \frac{1}{x}$$. 6. **Rewrite the entire expression:** $$\frac{y + 12}{x - 8} + \frac{1}{x}$$. 7. **Find common denominator:** The common denominator is $$x(x - 8)$$. 8. **Rewrite each fraction with common denominator:** $$\frac{y + 12}{x - 8} = \frac{x(y + 12)}{x(x - 8)}$$ $$\frac{1}{x} = \frac{x - 8}{x(x - 8)}$$ 9. **Add the fractions:** $$\frac{x(y + 12) + (x - 8)}{x(x - 8)} = \frac{xy + 12x + x - 8}{x(x - 8)} = \frac{xy + 13x - 8}{x(x - 8)}$$. 10. **Rewrite denominator:** $$x(x - 8) = x^{2} - 8x$$. 11. **Check options:** - Option C numerator: $$xy^{2} + 13xy - 8y$$ (has extra $$y$$ factors, does not match) - Option D numerator: same as C, denominator different - Option A and B have more complicated denominators. 12. **Try factoring numerator:** Our numerator is $$xy + 13x - 8$$, which does not match any option exactly. 13. **Re-examine step 5:** We simplified $$\frac{y(x - 8)}{xy(x - 8)}$$ to $$\frac{1}{x}$$, but numerator is $$y(x - 8)$$, denominator is $$xy(x - 8)$$. Cancel $$x - 8$$: $$\frac{y}{xy} = \frac{1}{x}$$. 14. **Final simplified expression:** $$\frac{y + 12}{x - 8} + \frac{1}{x} = \frac{x(y + 12) + (x - 8)}{x(x - 8)} = \frac{xy + 12x + x - 8}{x^{2} - 8x} = \frac{xy + 13x - 8}{x^{2} - 8x}$$. 15. **Compare with options:** Option C denominator is $$x^{2}y - 8xy = xy(x - 8)$$, which is different from our denominator $$x^{2} - 8x$$. 16. **Multiply numerator and denominator by $$y$$ to match denominator:** Multiply numerator and denominator by $$y$$: $$\frac{y(xy + 13x - 8)}{y(x^{2} - 8x)} = \frac{xy^{2} + 13xy - 8y}{x^{2}y - 8xy}$$. This matches Option C exactly. **Final answer:** Option C