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

Firstly multiply through by (x+1)(2x+1) to get rid of the fractions and then expand the brackets. Then gather the terms on one side to get a quadratic equation (ie 3x^2 - x -2=0) and then simply solve this either by inspection or using the quadratic formula. We see that trying x=1 indeed gets us 0 hence we know that one root of the equation is (x-1) now all that's left is finding the other root which can be found by 'working backgrounds' ie what multiplies with x-1 to get 3x^2 - x -2. Doing this we obtain x=-2/3.