`(a + 1)/(a^2 + 5a) xx (a^2 - 25)/(a^2 - a - 20) div (a^2 - a - 2)/(a^2 + 2a - 8)` when simplified is equal to :

Options

A. `1`
B. `a `
C. `1/a`
D. `a^2`

Answer

To simplify the expression \((a + 1)/(a^2 + 5a) \times (a^2 - 25)/(a^2 - a - 20) \div (a^2 - a - 2)/(a^2 + 2a - 8)\), we will follow these steps:

### Step 1: Factor the polynomials
1. **Factor \(a^2 + 5a\)**:
\[ a^2 + 5a = a(a + 5) \]

2. **Factor \(a^2 - 25\)** (difference of squares):
\[ a^2 - 25 = (a + 5)(a - 5) \]

3. **Factor \(a^2 - a - 20\)**:
\[ a^2 - a - 20 = (a - 5)(a + 4) \]

4. **Factor \(a^2 - a - 2\)**:
\[ a^2 - a - 2 = (a - 2)(a + 1) \]

5. **Factor \(a^2 + 2a - 8\)**:
\[ a^2 + 2a - 8 = (a + 4)(a - 2) \]

### Step 2: Rewrite the expression with the factors
Now substituting the factors back into the expression:
\[ \frac{a + 1}{a(a + 5)} \times \frac{(a + 5)(a - 5)}{(a - 5)(a + 4)} \div \frac{(a - 2)(a + 1)}{(a + 4)(a - 2)} \]

### Step 3: Change division to multiplication
Change the division to multiplication by taking the reciprocal:
\[ \frac{a + 1}{a(a + 5)} \times \frac{(a + 5)(a - 5)}{(a - 5)(a + 4)} \times \frac{(a + 4)(a - 2)}{(a - 2)(a + 1)} \]

### Step 4: Cancel common factors
Now we can cancel the common factors:
- \(a + 1\) cancels with \(a + 1\)
- \(a + 5\) cancels with \(a + 5\)
- \(a - 5\) cancels with \(a - 5\)
- \(a - 2\) cancels with \(a - 2\)
- \(a + 4\) cancels with \(a + 4\)

After canceling, we are left with:
\[ \frac{1}{a} \]

### Final Answer
Thus, the simplified expression is:
\[ \frac{1}{a} \]

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