80. $$ \frac{x^{2}-x-2}{x^{2}-7 x+10} \div \frac{x^{2}-3 x-4}{40-3 x-x^{2}} $$

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UpStudy AI · Accepted Answer

To simplify the expression:

$$
\frac{x^{2} - x - 2}{x^{2} - 7x + 10} \div \frac{x^{2} - 3x - 4}{40 - 3x - x^{2}}
$$

we follow these steps:

1. **Rewrite the Division as Multiplication:**

$$
\frac{x^{2} - x - 2}{x^{2} - 7x + 10} \times \frac{40 - 3x - x^{2}}{x^{2} - 3x - 4}
$$

2. **Factor Each Polynomial:**

- **Factor $$x^{2} - x - 2$$:**
     $$
     x^{2} - x - 2 = (x - 2)(x + 1)
     $$
   
   - **Factor $$x^{2} - 7x + 10$$:**
     $$
     x^{2} - 7x + 10 = (x - 2)(x - 5)
     $$
   
   - **Factor $$x^{2} - 3x - 4$$:**
     $$
     x^{2} - 3x - 4 = (x - 4)(x + 1)
     $$
   
   - **Factor $$40 - 3x - x^{2}$$:**
     $$
     40 - 3x - x^{2} = -x^{2} - 3x + 40 = -(x^{2} + 3x - 40) = -(x + 8)(x - 5)
     $$

3. **Substitute the Factors Back into the Expression:**

$$
\frac{(x - 2)(x + 1)}{(x - 2)(x - 5)} \times \frac{-(x + 8)(x - 5)}{(x - 4)(x + 1)}
$$

4. **Cancel Common Factors:**

- $$ (x - 2) $$ cancels out.
- $$ (x + 1) $$ cancels out.
- $$ (x - 5) $$ cancels out.

This leaves:

$$
\frac{- (x + 8)}{(x - 4)}
$$

5. **Final Simplified Form:**

$$
\frac{-(x + 8)}{x - 4}
$$

**Answer:**

After simplifying, the expression is –(x + 8) divided by (x – 4). Thus,

−(x+8)/(x−4)
The simplified form of the expression is $$-\frac{x + 8}{x - 4}$$.

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