To solve this equation, we will first distribute and simplify on both sides:

2(x+3) - 3(x+2) = 5 - 4(x+1)
2x + 6 - 3x - 6 = 5 - 4x - 4

Now, we will combine like terms:

Next, we will move all terms involving x to one side of the equation:

4x - x = -3
3x = -3

Finally, we will solve for x by dividing by 3:

x = -1

Therefore, the solution to the equation 2(x+3) - 3(x+2) = 5 - 4(x+1) is x = -1.