1. **State the problem:** Solve for $x$ in the equation $2(x-3)+3(x+1)=4(x-2)$.\n\n2. **Write down the equation:** $$2(x-3)+3(x+1)=4(x-2)$$\n\n3. **Distribute the constants:** Multiply each term inside the parentheses by the number outside.\n$$2 \times x - 2 \times 3 + 3 \times x + 3 \times 1 = 4 \times x - 4 \times 2$$\nWhich simplifies to:\n$$2x - 6 + 3x + 3 = 4x - 8$$\n\n4. **Combine like terms on the left side:**\n$$2x + 3x - 6 + 3 = 4x - 8$$\n$$5x - 3 = 4x - 8$$\n\n5. **Isolate variable terms on one side:** Subtract $4x$ from both sides.\n$$5x - 4x - 3 = -8$$\n$$x - 3 = -8$$\n\n6. **Solve for $x$:** Add 3 to both sides.\n$$x = -8 + 3$$\n$$x = -5$$\n\n**Final answer:** $x = -5$