1. **State the problem:** Simplify the algebraic expression $3x - 9 - 6x + 12 + 4x$. 2. **Combine like terms:** Group the terms with $x$ and the constant terms separately. $$3x - 6x + 4x - 9 + 12$$ 3. **Simplify the $x$ terms:** $$\cancel{3x} - \cancel{6x} + 4x = (3 - 6 + 4)x = 1x = x$$ 4. **Simplify the constants:** $$-9 + 12 = 3$$ 5. **Write the simplified expression:** $$x + 3$$ **Final answer:** $x + 3$