To solve this equation, we first need to eliminate the fractions by cross multiplying:

(x-1)(x-2) = (x+3)(x+2)

Expanding both sides:
x^2 -2x -x + 2 = x^2 +2x +3x +6

Simplifying:
x^2 -3x + 2 = x^2 + 5x + 6

Now, we can combine like terms to solve for x:
-3x + 2 = 5x + 6
-3x - 5x = 6 - 2
-8x = 4
x = -1/2

Therefore, the solution to the equation is x = -1/2.