First, note that because this is a quadratic equation, there must be two roots. You might be tempted to divide both sides by (2x-1), and get rid of it; but this discards one of your roots.
(2x - 1) = 0 --> This must be true, because you're multiplying both sides by it. Therefore, either it is zero, OR both 3x + 5 = 0 and x + 2 = 0, which is impossible. Thus, x = 1/2 is your first root.
To get the second root, divide by 2x-1 to get:
3x + 5 = x + 2 --> x = -3/2, your second root
Alternatively, you could have multiplied the two sides out and combined like terms to get the quadratic formula:
6 x^2+7 x-5 = 2 x^2+3 x-2
4x^2 + 4x - 3 = 0
x = (b +/- sqrt(b^2 - 4ac) ) /2a
yields x = 1/2, -2/3