To simplify the given expression \(\frac{2\sqrt{5} + 10}{5 - 9}\), we will follow these steps:

### Step 1: Simplify the Denominator
First, simplify the denominator:

\[ 5 - 9 = -4 \]

### Step 2: Rewrite the Expression
Rewrite the entire expression with the simplified denominator:

\[ \frac{2\sqrt{5} + 10}{-4} \]

### Step 3: Distribute the Denominator
Distribute the denominator to each term in the numerator:

\[ \frac{2\sqrt{5}}{-4} + \frac{10}{-4} \]

### Step 4: Simplify Each Fraction
Simplify each fraction individually:

\[ \frac{2\sqrt{5}}{-4} = -\frac{\sqrt{5}}{2} \]

\[ \frac{10}{-4} = -\frac{5}{2} \]

### Step 5: Combine the Simplified Terms
Combine the simplified terms:

\[ -\frac{\sqrt{5}}{2} - \frac{5}{2} \]

Thus, the simplified expression is:

\[ -\frac{\sqrt{5}}{2} - \frac{5}{2} \]

Would you like more details or have any questions about this solution? Here are five questions you might ask next:

1. How do you distribute a denominator to each term in the numerator?
2. Can you explain how to simplify fractions with square roots?
3. What are some common mistakes to avoid when simplifying expressions?
4. How do you add and subtract fractions with different denominators?
5. Can you show another example of simplifying a similar expression with different values?

**Tip:** Always simplify the denominator first when working with fractions. This can make the rest of the process easier and more straightforward.

Learn how to simplify the fraction expression (2sqrt(5) + 10)/(5 - 9) step-by-step with detailed explanations. This problem involves fraction simplification and working with square roots, suitable for students in Grades 9-12.

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