I assume this is a proportion - one fraction that equals another fraction. Here is the equation with parentheses to help avoid confusion.
(z + 4) / (z + 5) = -1 / (z + 5)
Use the Means Extremes Property. According to the Means Extremes Property...
If a/b = c/d, the ad = bc.
(z + 4)(z + 5) = -1(z + 5)
z^2 + 9z + 20 = -1z + -5
z^2 + 10z + 25 = 0
Factor the trinomial.
(z + 5)^2 = 0
z + 5 = 0
z = -5
However, z `!=` -5 because that would cause the denominators of the fractions to equal 0, which is impossible.
If this problem is not a proportion, the problem would be solved differently.
z + 4/z + 5 = -1/z + 5
Multiply both sides by z to eliminate the fractions.
z^2 + 4 + 5z = -1 + 5z
z^2 + 4 = -1
z^2 = -5
z = sqrt(-5)
You cannot square root a negative number, therefore there is no real solution.
Either way, the equation has no solution.
Is it possible you mistyped the equation, or forgot to insert parentheses?