Question: Solve the equation $$\frac{2x}{3x^2 - x+2} - \frac{7x}{3x^2 + 5x+2} = 1$$
I attempted to split the two quadratic equations into their roots, however, was unable to do so. Then I tried to solve by multiplying and removing the denominator, but the process became very cumbersome. Any easier way to do it?
Setting $A=3x^2-x+2$ gives you $$\begin{align}\frac{2x}{A}-\frac{7x}{A+6x}=1&\iff 2x(A+6x)-7xA=A(A+6x), A\not=0,A+6x\not =0\\&\iff A^2+11Ax-12x^2=0, A\not=0,A+6x\not =0\\&\iff (A+12x)(A-x)=0, A\not=0,A+6x\not =0\end{align}$$