## Long Division of Polynomials: (x^2 + 8x + 15) / (x + 5)

Long division of polynomials is a method for dividing two polynomials. It is similar to the long division of numbers, but we work with variables and coefficients.

In this example, we will divide the polynomial (x^2 + 8x + 15) by the polynomial (x + 5). Here's how to do it:

**1. Set up the division:**

```
_______
x + 5 | x^2 + 8x + 15
```

**2. Divide the leading terms:**

- The leading term of the divisor (x + 5) is 'x'.
- The leading term of the dividend (x^2 + 8x + 15) is 'x^2'.
- Divide x^2 by x, which gives us 'x'.
- Write 'x' above the x^2 term in the quotient.

```
x _______
x + 5 | x^2 + 8x + 15
```

**3. Multiply the quotient by the divisor:**

- Multiply 'x' (the quotient term) by (x + 5), which gives us 'x^2 + 5x'.

```
x _______
x + 5 | x^2 + 8x + 15
x^2 + 5x
```

**4. Subtract:**

- Subtract 'x^2 + 5x' from the dividend.

```
x _______
x + 5 | x^2 + 8x + 15
x^2 + 5x
-------
3x
```

**5. Bring down the next term:**

- Bring down the next term from the dividend, which is '+ 15'.

```
x _______
x + 5 | x^2 + 8x + 15
x^2 + 5x
-------
3x + 15
```

**6. Repeat steps 2-5:**

- Divide the leading term of the new dividend ('3x') by the leading term of the divisor ('x'), which gives us '3'.
- Write '3' next to 'x' in the quotient.
- Multiply '3' by (x + 5), which gives us '3x + 15'.
- Subtract '3x + 15' from the dividend.

```
x + 3
x + 5 | x^2 + 8x + 15
x^2 + 5x
-------
3x + 15
3x + 15
-------
0
```

**7. Result:**

Since the remainder is 0, the division is complete. Therefore:

**(x^2 + 8x + 15) / (x + 5) = x + 3**

**Important Note:** You can check your answer by multiplying the quotient (x + 3) by the divisor (x + 5). The result should be the original dividend (x^2 + 8x + 15).