https://file.aiquickdraw.com/mathbot/file/17266321042364iyb0yte.jpg

The expression shown is:

\[
\frac{a^2 - 4a - 5}{2a^2 - 50} \cdot \frac{2a - 2}{3a + 3}
\]

### Let's simplify the expression step by step:

1. **Factorize each term where possible**:

   - The numerator \(a^2 - 4a - 5\) can be factored as:
     \[
     a^2 - 4a - 5 = (a - 5)(a + 1)
     \]

   - The denominator \(2a^2 - 50\) can be factored by factoring out the common factor of 2:
     \[
     2a^2 - 50 = 2(a^2 - 25) = 2(a - 5)(a + 5)
     \]

   - The numerator \(2a - 2\) can be factored by factoring out the common factor of 2:
     \[
     2a - 2 = 2(a - 1)
     \]

   - The denominator \(3a + 3\) can be factored by factoring out the common factor of 3:
     \[
     3a + 3 = 3(a + 1)
     \]

2. **Rewrite the expression with the factored terms**:
   \[
   \frac{(a - 5)(a + 1)}{2(a - 5)(a + 5)} \cdot \frac{2(a - 1)}{3(a + 1)}
   \]

3. **Cancel common factors**:

   - The factor \(a - 5\) appears in both the numerator and denominator, so it cancels out.
   - The factor \(a + 1\) appears in both the numerator and denominator, so it cancels out.
   - The factor of 2 in the second fraction numerator cancels with the factor of 2 in the first fraction denominator.

   After canceling, we are left with:
   \[
   \frac{a - 1}{3(a + 5)}
   \]

Thus, the simplified expression is:

\[
\frac{a - 1}{3(a + 5)}
\]

Would you like more details or have any questions about this? 

### Related Questions:
1. What happens if \(a = -5\) in the simplified expression?
2. How can you solve this expression for \(a\) if it equals a particular value?
3. What are some applications of factoring quadratic expressions like these?
4. Could you simplify other algebraic expressions similarly?
5. How would the expression change if the coefficients in the numerator or denominator changed?

### Tip:
Always check for common factors between numerators and denominators before multiplying or simplifying! This helps reduce the complexity early on.

Simplify the expression: 

(a² - 4a - 5) / (2a² - 50) * (2a - 2) / (3a + 3)

This guide explores how to simplify complex algebraic fractions by factoring quadratic expressions and reducing common factors. It provides step-by-step solutions, relevant formulas, and applications of the Factor Theorem, making it ideal for high school students.

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