## Synthetic Division: A Step-by-Step Guide to Dividing (x³ + 5x² - x - 9) by (x + 2)

Synthetic division is a shortcut method for dividing polynomials, especially when the divisor is a linear expression of the form (x - a). In this case, we'll divide (x³ + 5x² - x - 9) by (x + 2).

**Step 1: Set up the problem**

- Write the coefficients of the dividend (x³ + 5x² - x - 9) in a row: 1 5 -1 -9.
- Write the constant term of the divisor (x + 2) with its sign changed: -2.
- Draw a horizontal line below the coefficients.

```
_________________
-2 | 1 5 -1 -9
------------------
```

**Step 2: Bring down the first coefficient**

- Bring down the first coefficient (1) below the line.

```
_________________
-2 | 1 5 -1 -9
------------------
1
```

**Step 3: Multiply and add**

- Multiply the number you just brought down (1) by the divisor's constant term (-2) and write the product ( -2 ) under the next coefficient (5).
- Add the numbers in the column (5 + -2 = 3).

```
_________________
-2 | 1 5 -1 -9
------------------
1 3
```

**Step 4: Repeat the multiplication and addition process**

- Multiply the new number (3) by the divisor's constant term (-2) and write the product (-6) under the next coefficient (-1).
- Add the numbers in the column (-1 + -6 = -7).

```
_________________
-2 | 1 5 -1 -9
------------------
1 3 -7
```

**Step 5: Continue the process until you reach the last coefficient**

- Multiply the new number (-7) by the divisor's constant term (-2) and write the product (14) under the last coefficient (-9).
- Add the numbers in the column (-9 + 14 = 5).

```
_________________
-2 | 1 5 -1 -9
------------------
1 3 -7 5
```

**Step 6: Interpret the results**

- The numbers below the line represent the coefficients of the quotient, starting from the highest power of x.
- The last number (5) represents the remainder.

Therefore, the result of the division (x³ + 5x² - x - 9) / (x + 2) is:

**x² + 3x - 7 + 5/(x + 2)**