1. **Stating the problem:** Factorize the expression $$5a^6 b^2 - 15a^3 b^5 + 35a^4 b^4$$. 2. **Identify the greatest common factor (GCF):** - Coefficients: GCF of 5, 15, and 35 is 5. - For $a$: minimum power is $a^3$ (since powers are 6, 3, and 4). - For $b$: minimum power is $b^2$ (since powers are 2, 5, and 4). So, GCF is $$5a^3 b^2$$. 3. **Factor out the GCF:** $$5a^3 b^2 (\frac{5a^6 b^2}{5a^3 b^2} - \frac{15a^3 b^5}{5a^3 b^2} + \frac{35a^4 b^4}{5a^3 b^2})$$ Simplify each term inside the parentheses: - $$\frac{5a^6 b^2}{5a^3 b^2} = a^{6-3} b^{2-2} = a^3$$ - $$\frac{15a^3 b^5}{5a^3 b^2} = 3 b^{5-2} = 3 b^3$$ - $$\frac{35a^4 b^4}{5a^3 b^2} = 7 a^{4-3} b^{4-2} = 7 a b^2$$ 4. **Rewrite the factored expression:** $$5a^3 b^2 (a^3 - 3 b^3 + 7 a b^2)$$ 5. **Conclusion:** The correct factorization matches option c. **Final answer:** $$5a^3 b^2 (a^3 - 3 b^3 + 7 a b^2)$$