1. **State the problem:** Simplify the expression $$f = \frac{3ab^2}{5xy} + \frac{12ab - 6a}{x^2 y + 2xy^2}$$. 2. **Identify the denominators:** The denominators are $$5xy$$ and $$x^2 y + 2xy^2$$. 3. **Factor the second denominator:** $$x^2 y + 2xy^2 = xy(x + 2y)$$. 4. **Rewrite the expression with factored denominator:** $$f = \frac{3ab^2}{5xy} + \frac{12ab - 6a}{xy(x + 2y)}$$. 5. **Factor the numerator of the second fraction:** $$12ab - 6a = 6a(2b - 1)$$. 6. **Rewrite the expression:** $$f = \frac{3ab^2}{5xy} + \frac{6a(2b - 1)}{xy(x + 2y)}$$. 7. **Find the least common denominator (LCD):** The denominators are $$5xy$$ and $$xy(x + 2y)$$. The LCD is $$5xy(x + 2y)$$. 8. **Rewrite each fraction with the LCD:** $$\frac{3ab^2}{5xy} = \frac{3ab^2 \cdot (x + 2y)}{5xy(x + 2y)}$$ $$\frac{6a(2b - 1)}{xy(x + 2y)} = \frac{6a(2b - 1) \cdot 5}{5xy(x + 2y)}$$ 9. **Multiply numerators:** $$3ab^2(x + 2y) = 3ab^2 x + 6ab^2 y$$ $$6a(2b - 1) \cdot 5 = 30a(2b - 1) = 60ab - 30a$$ 10. **Combine the fractions:** $$f = \frac{3ab^2 x + 6ab^2 y + 60ab - 30a}{5xy(x + 2y)}$$. 11. **Factor numerator if possible:** Group terms: $$3ab^2 x + 6ab^2 y + 60ab - 30a = 3a b^2 x + 6a b^2 y + 60a b - 30a$$ Factor out $$3a$$: $$3a(b^2 x + 2 b^2 y + 20 b - 10)$$ No further simple factorization is apparent. 12. **Final simplified form:** $$\boxed{f = \frac{3a(b^2 x + 2 b^2 y + 20 b - 10)}{5xy(x + 2y)}}$$