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?

We have to solve (z+4)/(z+5) = -1/(z+5) for z

(z+4)/(z+5) = -1/(z+5)

=> z + 4 = -1

=> z = -5

but at z = -5, (z + 5) = 0

As the denominator cannot be equal to zero, z cannot equal -5.

**The equation does not have a solution.**

(z+4)/(z+5)=-1/(z+5)

This is the equation

multiply z+5 to both sides, adding a restriction z could not be -5

z+4=-1

z=-5

the restriction says that z could not be -5

the answer is DNE (does not exist)

another way of doing this is to move the -1/(z+5) to the left side

becoming

(z+4+1)/(z+5)=0

(z+5)/(z+5)=0

1=0

This is NEVER true, so the answer is DNE

**The solution to this equation is DNE(Does not exist)**