Solve the equation $|2x^2+x-1|=|x^2+4x+1|$ Find the sum of all the solutions of the equation  $|2x^2+x-1|=|x^2+4x+1|$
Though I tried to solve it in desmos.com and getting the requisite answer but while solving it manually it is getting very lengthy. 
I tried to construct the two parabola and mirror image the region below y axis but still getting it is getting complicated. 

Is there any easy method to solve it and get the sum of all the solutions ?


The expressions between the absolute value bars have the same or opposite signs. Hence there are two independent cases (by addition and subtraction):
$$3x^2+5x=0$$ and $$x^2-3x-2=0.$$
Then by Vieta,
$$-\frac53+3.$$

For complete rigor, one should show that no root is repeated. This is true, because the polynomials have no double root, and their $\text{gcd}$ is $1$.
