To simplify the expression \((x^2 - x - 30)/(36 - 6x)\), we'll start by factoring both the numerator and the denominator.

### Step 1: Factor the Numerator

The numerator is \(x^2 - x - 30\). We need to find two numbers that multiply to \(-30\) (the constant term) and add up to \(-1\) (the coefficient of \(x\)).

The factors of \(-30\) that add up to \(-1\) are \(-6\) and \(5\). So, we can factor the numerator as:

\[
x^2 - x - 30 = (x - 6)(x + 5)
\]

### Step 2: Factor the Denominator

The denominator is \(36 - 6x\). This can be factored by taking out the common factor of \(6\):

\[
36 - 6x = 6(6 - x)
\]

### Step 3: Rewrite the Expression with Factored Terms

Now, substitute the factored forms back into the original expression:

\[
\frac{(x - 6)(x + 5)}{6(6 - x)}
\]

### Step 4: Simplify the Expression

Notice that \(6 - x = -(x - 6)\). We can rewrite the denominator using this relationship:

\[
\frac{(x - 6)(x + 5)}{6(-(x - 6))}
\]

This simplifies to:

\[
- \frac{(x - 6)(x + 5)}{6(x - 6)}
\]

Now, we can cancel out the common factor \((x - 6)\), assuming \(x \neq 6\):

\[
- \frac{x + 5}{6}
\]

### Final Simplified Expression

The simplified expression is:

\[
- \frac{x + 5}{6}, \quad \text{where } x \neq 6
\]

### Conclusion

The expression \((x^2 - x - 30)/(36 - 6x)\) simplifies to \(-\frac{x + 5}{6}\), with the condition that \(x \neq 6\) to avoid division by zero.

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Do you have any questions or would you like further details?

### Related Questions:

1. How do you factor quadratic expressions in general?
2. What are the conditions for canceling terms in a fraction?
3. How can you rewrite expressions when dealing with negative signs?
4. What is the process of factoring polynomials with higher degrees?
5. How does changing the value of \(x\) affect the expression?

**Tip:** Always check for any restrictions on the variables (like \(x \neq 6\) here) to avoid division by zero when simplifying expressions.

Learn how to simplify the rational expression (x^2 - x - 30)/(36 - 6x) with a detailed step-by-step solution. This problem covers key algebraic concepts, including factoring quadratic expressions and simplifying rational expressions. Suitable for students in Grades 9-10.

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