1. **State the problem:** Simplify or factor the expression $10a^3b^4 - 15a^4b^6 + 20a^2b^7$. 2. **Identify the common factors:** Look for the greatest common factor (GCF) in all terms. 3. The coefficients are 10, -15, and 20. The GCF of 10, 15, and 20 is 5. 4. For the variables, find the lowest powers of $a$ and $b$ present in all terms: - For $a$: powers are 3, 4, and 2, so the lowest is 2. - For $b$: powers are 4, 6, and 7, so the lowest is 4. 5. Therefore, the GCF is $5a^2b^4$. 6. **Factor out the GCF:** $$10a^3b^4 - 15a^4b^6 + 20a^2b^7 = 5a^2b^4\left(\frac{10a^3b^4}{5a^2b^4} - \frac{15a^4b^6}{5a^2b^4} + \frac{20a^2b^7}{5a^2b^4}\right)$$ 7. Simplify inside the parentheses: $$= 5a^2b^4\left(\cancel{\frac{10}{5}}a^{3-2}b^{4-4} - \cancel{\frac{15}{5}}a^{4-2}b^{6-4} + \cancel{\frac{20}{5}}a^{2-2}b^{7-4}\right)$$ $$= 5a^2b^4(2a^1b^0 - 3a^2b^2 + 4a^0b^3)$$ 8. Simplify powers of 0: $$= 5a^2b^4(2a - 3a^2b^2 + 4b^3)$$ **Final answer:** $$\boxed{5a^2b^4(2a - 3a^2b^2 + 4b^3)}$$