1. **State the problem:** Simplify the expression $$-\left[a^2 \cdot (a^2 - 2a^2) + a - 3\right] + \left[(-2a^2)^2 + (a + 1)^2 - \frac{6a^3b}{6ab}\right] - (4 + a)$$. 2. **Simplify inside the first bracket:** $$a^2 \cdot (a^2 - 2a^2) = a^2 \cdot (-a^2) = -a^4$$ So the first bracket becomes: $$-a^4 + a - 3$$ 3. **Apply the negative sign outside the first bracket:** $$-\left(-a^4 + a - 3\right) = a^4 - a + 3$$ 4. **Simplify the second bracket:** - Calculate each term: $$(-2a^2)^2 = (-2)^2 \cdot (a^2)^2 = 4a^4$$ $$ (a + 1)^2 = a^2 + 2a + 1$$ $$ \frac{6a^3b}{6ab} = \cancel{6}a^{3-1}b^{1-1} / \cancel{6} = a^2$$ - Substitute back: $$4a^4 + (a^2 + 2a + 1) - a^2 = 4a^4 + 2a + 1$$ 5. **Combine all parts:** $$a^4 - a + 3 + 4a^4 + 2a + 1 - (4 + a)$$ 6. **Simplify the last subtraction:** $$-(4 + a) = -4 - a$$ 7. **Sum all terms:** $$a^4 + 4a^4 = 5a^4$$ $$-a + 2a - a = 0$$ $$3 + 1 - 4 = 0$$ 8. **Final simplified expression:** $$5a^4$$ **Answer:** $$5a^4$$