Question 151771
Note that 
{{{z^2 + 9z + 20 = (z + 4)(z + 5)}}}
{{{z^2 - z - 2 = (z - 2)(z + 1)}}}

So the original equation becomes:
{{{(z + 3)/(z + 1) = ((z + 4)(z + 5))/((z - 2)(z + 1))}}}
Multiplying both sides by (z - 2)(z + 1), we have
(z + 3)(z - 2) = (z + 4)(z + 5)
Expanding, we have
{{{z^2 + z - 6 = z^2 + 9z + 20}}}
Simplifying, we have
-26 = 8z
z = -26/8 = -13/4

CHECK:
Left hand side = (-13/4) + 3)/(-13/4 + 1) = 1/9
Right hand side = [(-13/4)^2 + 9(-13/4) + 20]/[(-13/4)^2 - (-13/4) - 2]= 1/9
So z = -13/4 is the root of the original equation.