1. **State the problem:** Solve for $x$ in the equation $2(x-3)+3(x+1)=4(x-2)$. 2. **Write down the equation:** $$2(x-3)+3(x+1)=4(x-2)$$ 3. **Apply the distributive property:** $$2x - 6 + 3x + 3 = 4x - 8$$ 4. **Combine like terms on the left side:** $$5x - 3 = 4x - 8$$ 5. **Isolate variable terms on one side:** Subtract $4x$ from both sides: $$5x - 4x - 3 = -8$$ $$x - 3 = -8$$ 6. **Isolate $x$ by adding 3 to both sides:** $$x = -8 + 3$$ $$x = -5$$ **Final answer:** $$x = -5$$