1. Problem (a): Simplify the expression $2a(-4a^2 + 3a - 2) - 5(4a - a^2 - 2a^3)$. 2. Use the distributive property: $a(b + c) = ab + ac$. 3. Distribute $2a$ over $(-4a^2 + 3a - 2)$: $$2a \times -4a^2 = -8a^3$$ $$2a \times 3a = 6a^2$$ $$2a \times -2 = -4a$$ So, $2a(-4a^2 + 3a - 2) = -8a^3 + 6a^2 - 4a$. 4. Distribute $-5$ over $(4a - a^2 - 2a^3)$: $$-5 \times 4a = -20a$$ $$-5 \times -a^2 = +5a^2$$ $$-5 \times -2a^3 = +10a^3$$ So, $-5(4a - a^2 - 2a^3) = -20a + 5a^2 + 10a^3$. 5. Combine the two results: $$-8a^3 + 6a^2 - 4a - 20a + 5a^2 + 10a^3$$ 6. Group like terms: $$(-8a^3 + 10a^3) + (6a^2 + 5a^2) + (-4a - 20a)$$ 7. Simplify each group: $$2a^3 + 11a^2 - 24a$$ Final answer: $$\boxed{2a^3 + 11a^2 - 24a}$$