## Long Division of Polynomials: (5x^4 - 2x^3 - 7x^2 - 39) / (x^2 + 2x - 4)

This article will guide you through the process of performing long division with polynomials, specifically focusing on the example: (5x^4 - 2x^3 - 7x^2 - 39) / (x^2 + 2x - 4).

### Step 1: Setting Up the Division

Begin by writing the problem in a long division format:

```
________
x^2 + 2x - 4 | 5x^4 - 2x^3 - 7x^2 - 39
```

### Step 2: Dividing the Leading Terms

Focus on the leading terms of both the divisor (x^2 + 2x - 4) and the dividend (5x^4 - 2x^3 - 7x^2 - 39).

- Ask yourself: "What do I need to multiply x^2 by to get 5x^4?"
- The answer is 5x^2. Write this above the line in the quotient area.

```
5x^2
x^2 + 2x - 4 | 5x^4 - 2x^3 - 7x^2 - 39
```

### Step 3: Multiply and Subtract

- Multiply the entire divisor (x^2 + 2x - 4) by the term you just wrote in the quotient (5x^2).
- Write the result below the dividend, aligning like terms.
- Subtract the result from the dividend.

```
5x^2
x^2 + 2x - 4 | 5x^4 - 2x^3 - 7x^2 - 39
-(5x^4 + 10x^3 - 20x^2)
-----------------------
-12x^3 + 13x^2 - 39
```

### Step 4: Repeat the Process

- Bring down the next term (-39) from the dividend.
- Focus on the new leading term of the dividend (-12x^3) and the leading term of the divisor (x^2).
- Ask: "What do I multiply x^2 by to get -12x^3?"
- The answer is -12x. Write this next to 5x^2 in the quotient.

```
5x^2 - 12x
x^2 + 2x - 4 | 5x^4 - 2x^3 - 7x^2 - 39
-(5x^4 + 10x^3 - 20x^2)
-----------------------
-12x^3 + 13x^2 - 39
-(-12x^3 - 24x^2 + 48x)
-----------------------
37x^2 - 48x - 39
```

### Step 5: Continue until the Degree is Lower

Keep repeating steps 2-4. Continue dividing until the degree of the remaining polynomial in the dividend is lower than the degree of the divisor (x^2 + 2x - 4).

```
5x^2 - 12x + 37
x^2 + 2x - 4 | 5x^4 - 2x^3 - 7x^2 - 39
-(5x^4 + 10x^3 - 20x^2)
-----------------------
-12x^3 + 13x^2 - 39
-(-12x^3 - 24x^2 + 48x)
-----------------------
37x^2 - 48x - 39
-(37x^2 + 74x - 148)
--------------------
-122x + 109
```

### Step 6: Express the Remainder

We stop here because the degree of the remaining polynomial (-122x + 109) is less than the degree of the divisor (x^2 + 2x - 4). This remaining polynomial is the remainder.

### Step 7: The Final Answer

The final result of the long division is expressed as:

**(5x^4 - 2x^3 - 7x^2 - 39) / (x^2 + 2x - 4) = 5x^2 - 12x + 37 + (-122x + 109) / (x^2 + 2x - 4)**

This means the quotient is **5x^2 - 12x + 37** and the remainder is **-122x + 109**.