We need to simplify the following expression:

$\left( \frac{4z}{-5 - z} \right) - \left( \frac{-4z}{125 - 5z^2} \right)$

Step 1: Simplify each fraction individually

First Fraction: $\frac{4z}{-5 - z}$

This fraction cannot be simplified further, so we leave it as:

$\frac{4z}{-5 - z}$

Second Fraction: $\frac{-4z}{125 - 5z^2}$

Factor the denominator $125 - 5z^2$ using the difference of squares:

$125 - 5z^2 = 5(25 - z^2) = 5(5 - z)(5 + z)$

Now, the second fraction becomes:

$\frac{-4z}{5(5 - z)(5 + z)}$

Step 2: Combine the two terms under a common denominator

To combine the two fractions, we need a common denominator. The denominator of the first fraction is $(-5 - z)$, which is equivalent to $-(z + 5)$. So the common denominator between the two fractions is $5(z + 5)(z - 5)$.

Rewrite both fractions with this common denominator:

1. The first fraction becomes: $\frac{4z}{-(z + 5)} = \frac{-4z}{z + 5}$ To get the common denominator, multiply the numerator and denominator by $5 - z$: $\frac{-4z(5 - z)}{(z + 5)(5 - z)} = \frac{-4z(5 - z)}{(z + 5)(z - 5)}$

2. The second fraction becomes: $\frac{-4z}{5(z + 5)(z - 5)}$

Step 3: Combine the two fractions

Now that both fractions have the same denominator, combine them:

$\frac{-4z(5 - z)}{(z + 5)(z - 5)} - \frac{-4z}{5(z + 5)(z - 5)}$

Factor out $-4z$ from both numerators:

$\frac{-4z \left[ (5 - z) + \frac{1}{5} \right]}{5(z + 5)(z - 5)}$

Simplify the expression inside the brackets:

$(5 - z) + \frac{1}{5} = \frac{25 - 5z + 1}{5} = \frac{26 - 5z}{5}$

Now, the expression becomes:

$\frac{-4z \cdot \frac{26 - 5z}{5}}{5(z + 5)(z - 5)}$

Step 4: Simplify the final expression

Multiply the numerators:

$\frac{-4z(26 - 5z)}{25(z + 5)(z - 5)}$

This is the simplified form of the expression.

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Would you like a detailed explanation of any step or further breakdowns? Here are some related questions you might find useful:

1. How do you factor polynomials with higher powers of $z$?
2. What is the difference of squares, and how is it used in simplifications?
3. How can you simplify complex rational expressions with multiple terms?
4. What strategies work best for combining fractions with different denominators?
5. How can you use factoring in simplifying expressions?

Tip: Always look for factoring opportunities, especially with quadratic terms like $z^2$, to make simplifying fractions easier!