Start by multiplying both sides by 4*3*2*1=24, but we'll group the factors carefully:
(4*(3z+1))*(3*(4z+1))*(2*(6z+1))*(1*(12z+1)) = 24*2
(12z+4) * (12z+3) * (12z+2) * (12z+1) = 48
On the left multiply the first and fourth factors and similarly the second and third factors:
((12z+4)*(12z+1)) * ((12z+2)*(12z+3)) = 48
(144z^2+5z+4) * (144z^2+5z+6) = 48
Now, the left side is very close to being factorable as a difference of squares:
(144z^2+5z+5 - 1) * (144z^2+5z+5 + 1) = 48
(144z^2+5z+5)^2 - 1^2 = 48
(144z^2+5z+5)^2 = 49
Now this is fairly simple to break into two quadratics:
144z^2+5z+5 = 7 OR 144z^2+5z+5 = -7
Then two uses of the quadratic formula gives us four roots of:
z = (-5 +/- i*sqrt(23)/24 OR (-5 +/- sqrt(33)/24.
Solution (An easier way)
