Solve x for (x)/(4x-1) = (6x+5)/(12x+31)

First, we must cross multiply to remove the fractions, to do this, we multiply each numerator by the opposite denominator: (x)(12x+31) = (6x+5)(4x-1). Then, we multiply out the brackets to get: 12x² + 31x = 24x² +20x -6x -5, which simplifies to 12x² + 31x = 24x² +14x -5. Now we must rearrange to form a quadratic equation that is equal to zero in order to factorise: so take -12x² to both sides: 31x = 12x² +14x-5, and take -31x from both sides: 0=12x²-17x-5. Now we must factorise in order to find the solutions to x. For a quadratic like this, with a coefficient of x² greater than 1, we must first multiply 12 by 5 = 60. Now look for the factors of 60 that we can use to make our coefficient of x which is -17, in this case +3 and -20. We must also look at out factors of 12, which are 1 and 12, 2 and 6, 3 and 4. Only 3 and 4 can are also factors of 3 and 20. So our brackets become (4x....) and (3x....).Divide -20 by 4 which gives -5, and +3 by 3 which gives +1. Our brackets are now (4x+1)(3x-5)=0, the -5 and +1 must placed in the opposite bracket so if we were to multiply them out, we would get -20x and +3x. We can now equate each bracket to 0, (4x+1)=0 and (3x+5)=0, and solve through rearranging; x=-1/4 and x=5/3 are our answers.