Simplify the rational expression. Assume that variable z has no value which results in the denominator with a value of zero.
StartFraction 5 z squared plus 25 z plus 30 Over negative 10 z minus 30 EndFraction

To simplify the rational expression 

\[
\frac{5z^2 + 25z + 30}{-10z - 30},
\]

we can follow these steps:

1. **Factor the numerator and denominator:**
   - Numerator: \( 5z^2 + 25z + 30 \)
   - Denominator: \( -10z - 30 \)

2. **Factor out common factors:**
   - In the numerator, factor out 5:
     \[
     5z^2 + 25z + 30 = 5(z^2 + 5z + 6)
     \]
   - In the denominator, factor out \(-10\):
     \[
     -10z - 30 = -10(z + 3)
     \]

3. **Factor the quadratic expression in the numerator:**
   - Factor \( z^2 + 5z + 6 \):
     \[
     z^2 + 5z + 6 = (z + 2)(z + 3)
     \]

   So, the numerator becomes:
     \[
     5(z + 2)(z + 3)
     \]

   The expression now looks like:
     \[
     \frac{5(z + 2)(z + 3)}{-10(z + 3)}
     \]

4. **Simplify the expression:**
   - Cancel out the common factor \( (z + 3) \) in the numerator and denominator:
     \[
     \frac{5(z + 2) \cancel{(z + 3)}}{-10 \cancel{(z + 3)}} = \frac{5(z + 2)}{-10}
     \]

   - Simplify the fraction:
     \[
     \frac{5(z + 2)}{-10} = \frac{5}{-10}(z + 2) = -\frac{1}{2}(z + 2)
     \]

Therefore, the simplified form of the rational expression is:

\[
-\frac{1}{2}(z + 2)
\]

Would you like further details or have any questions?

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Here are some related questions you might find interesting:

1. How do you factor a quadratic expression like \( z^2 + 5z + 6 \)?
2. What happens if \( z = -3 \) in the original expression?
3. Can you explain how to simplify rational expressions with different types of denominators?
4. How do you simplify an expression when the numerator is not factorable?
5. What are the rules for canceling out terms in a rational expression?

**Tip:** Always check for common factors in both the numerator and denominator before simplifying a rational expression.

Learn how to simplify the rational expression 5z^2 + 25z + 30 / -10z - 30 with a detailed step-by-step solution. This problem covers key algebraic concepts, including rational expressions, factoring, and simplification. Suitable for students in Grades 9-12.

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