Divide `z (5z^2-80)` by `5z(z+4)`

Options

A. `z+4`
B. `z-5`
C. `z-4`
D. `z+5`

Explanation

To solve the problem of dividing \( z(5z^2 - 80) \) by \( 5z(z + 4) \), we will follow these steps:

### Step 1: Factor the numerator
The numerator is \( z(5z^2 - 80) \). We can factor out the common term from \( 5z^2 - 80 \).

1. **Factor out 5** from \( 5z^2 - 80 \):
\[ 5z^2 - 80 = 5(z^2 - 16) \]

2. **Recognize the difference of squares** in \( z^2 - 16 \):
\[ z^2 - 16 = (z + 4)(z - 4) \]

So, we can rewrite the numerator:
\[ z(5z^2 - 80) = z \cdot 5(z^2 - 16) = z \cdot 5(z + 4)(z - 4) \]

### Step 2: Rewrite the expression
Now, we can rewrite the entire expression we want to divide:
\[ \frac{z(5(z + 4)(z - 4))}{5z(z + 4)} \]

### Step 3: Cancel common factors
Now, we can simplify the expression by canceling out the common factors in the numerator and denominator.

1. **Cancel \( 5z \)** from both the numerator and the denominator:
\[ \frac{z \cdot 5(z + 4)(z - 4)}{5z(z + 4)} = \frac{(z - 4)}{1} \]

2. **Cancel \( (z + 4) \)** from both the numerator and the denominator:
\[ = z - 4 \]

### Final Answer
Thus, the result of dividing \( z(5z^2 - 80) \) by \( 5z(z + 4) \) is:
\[ \boxed{z - 4} \]

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