## Long Division of Polynomials: (x^3+x^2+3x-4)/(x^2+2x+1)

Long division of polynomials is a process used to divide a polynomial by another polynomial of a lower degree. Here's how to perform the long division of (x^3+x^2+3x-4) by (x^2+2x+1):

**1. Set up the Division:**

Write the problem in the standard long division format:

```
_________
x^2+2x+1 | x^3 + x^2 + 3x - 4
```

**2. Divide the Leading Terms:**

Divide the leading term of the dividend (x^3) by the leading term of the divisor (x^2). This gives you x. Write this above the division line, aligned with the x^3 term.

```
x
x^2+2x+1 | x^3 + x^2 + 3x - 4
```

**3. Multiply the Divisor:**

Multiply the divisor (x^2+2x+1) by the term you just wrote (x). Write the result below the dividend.

```
x
x^2+2x+1 | x^3 + x^2 + 3x - 4
x^3 + 2x^2 + x
```

**4. Subtract:**

Subtract the line you just wrote from the dividend.

```
x
x^2+2x+1 | x^3 + x^2 + 3x - 4
x^3 + 2x^2 + x
-------------
-x^2 + 2x - 4
```

**5. Bring Down the Next Term:**

Bring down the next term from the dividend (-4).

```
x
x^2+2x+1 | x^3 + x^2 + 3x - 4
x^3 + 2x^2 + x
-------------
-x^2 + 2x - 4
```

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

Now focus on the new dividend (-x^2 + 2x - 4).

- Divide the leading term of the new dividend (-x^2) by the leading term of the divisor (x^2). This gives you -1. Write this above the division line, aligned with the x^2 term.

```
x - 1
x^2+2x+1 | x^3 + x^2 + 3x - 4
x^3 + 2x^2 + x
-------------
-x^2 + 2x - 4
```

- Multiply the divisor (x^2+2x+1) by -1. Write the result below the new dividend.

```
x - 1
x^2+2x+1 | x^3 + x^2 + 3x - 4
x^3 + 2x^2 + x
-------------
-x^2 + 2x - 4
-x^2 - 2x - 1
```

- Subtract.

```
x - 1
x^2+2x+1 | x^3 + x^2 + 3x - 4
x^3 + 2x^2 + x
-------------
-x^2 + 2x - 4
-x^2 - 2x - 1
-------------
4x - 3
```

**7. Stop when the Degree of the Remainder is Less than the Degree of the Divisor:**

The degree of the remainder (4x-3) is 1, which is less than the degree of the divisor (x^2+2x+1). Therefore, we stop here.

**Result:**

The result of the long division is:

**(x^3+x^2+3x-4)/(x^2+2x+1) = x - 1 + (4x - 3)/(x^2+2x+1)**

This means the original polynomial can be expressed as:

**x^3 + x^2 + 3x - 4 = (x^2 + 2x + 1)(x - 1) + (4x - 3)**