1. **State the problem:** Simplify the expression $$\frac{4}{35} a^2 b - \frac{12}{5} a b + \frac{8}{15} a^2 b^3 - \frac{16}{25} a^3 b.$$ 2. **Group like terms:** Group terms with common variables and powers: $$\left(\frac{4}{35} a^2 b + \frac{8}{15} a^2 b^3\right) + \left(- \frac{12}{5} a b - \frac{16}{25} a^3 b\right).$$ 3. **Factor out common factors in each group:** - From the first group, factor out $$a^2 b$$: $$a^2 b \left(\frac{4}{35} + \frac{8}{15} b^2\right).$$ - From the second group, factor out $$a b$$: $$a b \left(- \frac{12}{5} - \frac{16}{25} a^2\right).$$ 4. **Simplify inside the parentheses:** - Find common denominators: For $$\frac{4}{35} + \frac{8}{15} b^2$$, the common denominator is 105: $$\frac{4}{35} = \frac{12}{105}, \quad \frac{8}{15} = \frac{56}{105}.$$ So, $$\frac{12}{105} + \frac{56}{105} b^2 = \frac{12 + 56 b^2}{105}.$$ - For $$- \frac{12}{5} - \frac{16}{25} a^2$$, the common denominator is 25: $$- \frac{12}{5} = - \frac{60}{25}.$$ So, $$- \frac{60}{25} - \frac{16}{25} a^2 = - \frac{60 + 16 a^2}{25}.$$ 5. **Rewrite the expression:** $$a^2 b \cdot \frac{12 + 56 b^2}{105} + a b \cdot \left(- \frac{60 + 16 a^2}{25}\right).$$ 6. **Final simplified form:** $$\frac{a^2 b (12 + 56 b^2)}{105} - \frac{a b (60 + 16 a^2)}{25}.$$ This is the simplified expression with factored terms and common denominators combined. **Answer:** $$\boxed{\frac{a^2 b (12 + 56 b^2)}{105} - \frac{a b (60 + 16 a^2)}{25}}.$$