(2x + 3)/(x-4) - (2x - 8)/(2x + 1) = 1

Multiply through by both denominators, giving us the equation (2x + 3)(2x + 1) - (2x - 8)(x - 4) = (x-4)(2x +1). If we expand the brackets, we arrive at 4x^2 + 8x + 3 - (2x^2 -16x + 32) = 2x^2 -7x -4. If we simply we now have 2x^2 + 24x - 29 = 2x^2 - 7x -4. At this stage we can spot we can remove 2x^2 from both sides and rearrange so that we have all the x terms on one side and all the numbers on the other. We do this by adding 7x and 29 to both sides. This gives us 31x = 25, dividing through by 31 gives us x = 25/31, which is the solution.