solve the equation (z+4)/(z+5)=-1/(z+5) for the unknown variable z.

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.

Therefore...

(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?