1. **State the problem:** We need to perform the operations: From the sum of $3a^2 - ab - 2b^2$ and $2a^2 + 5ab - 3b^2$, subtract $a^2 - 3ab - 4b^2$. 2. **Write the expression:** $$\left(3a^2 - ab - 2b^2\right) + \left(2a^2 + 5ab - 3b^2\right) - \left(a^2 - 3ab - 4b^2\right)$$ 3. **Add the first two polynomials:** $$3a^2 - ab - 2b^2 + 2a^2 + 5ab - 3b^2 = (3a^2 + 2a^2) + (-ab + 5ab) + (-2b^2 - 3b^2)$$ $$= 5a^2 + 4ab - 5b^2$$ 4. **Subtract the third polynomial:** $$\left(5a^2 + 4ab - 5b^2\right) - \left(a^2 - 3ab - 4b^2\right) = 5a^2 + 4ab - 5b^2 - a^2 + 3ab + 4b^2$$ 5. **Combine like terms:** $$ (5a^2 - a^2) + (4ab + 3ab) + (-5b^2 + 4b^2) = 4a^2 + 7ab - b^2$$ **Final answer:** $$4a^2 + 7ab - b^2$$