1. Stating the problem: Simplify the expression $$2a(-3a)^3 + (-4a^2)^2 + 7a^2 - (-3a)(-2a) - a^2$$. 2. Recall the rules: - When raising a power to a power, multiply exponents. - Negative signs inside parentheses affect the sign of the term. - Multiply coefficients and variables separately. 3. Calculate each term: - $$(-3a)^3 = (-3)^3 \cdot a^3 = -27a^3$$ - So, $$2a(-3a)^3 = 2a \cdot (-27a^3) = 2 \cdot (-27) \cdot a \cdot a^3 = -54a^{4}$$ 4. Next term: - $$(-4a^2)^2 = (-4)^2 \cdot (a^2)^2 = 16a^{4}$$ 5. Third term is $$7a^2$$. 6. Fourth term: - $$-(-3a)(-2a) = -[(-3)(-2) a \cdot a] = -[6a^{2}] = -6a^{2}$$ 7. Fifth term is $$-a^2$$. 8. Now sum all terms: $$-54a^{4} + 16a^{4} + 7a^{2} - 6a^{2} - a^{2}$$ 9. Combine like terms: - For $$a^{4}$$: $$-54a^{4} + 16a^{4} = (-54 + 16)a^{4} = -38a^{4}$$ - For $$a^{2}$$: $$7a^{2} - 6a^{2} - a^{2} = (7 - 6 - 1)a^{2} = 0$$ 10. Final simplified expression: $$\boxed{-38a^{4}}$$