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

This article will guide you through the process of dividing the polynomial **x^3 - 4x^2 + 3x + 2** by the binomial **x + 2** using long division.

### Understanding Long Division with Polynomials

Long division with polynomials works similarly to long division with numbers. We follow these steps:

**Set up the division:**Arrange the dividend (x^3 - 4x^2 + 3x + 2) and divisor (x + 2) in the traditional long division format.**Divide the leading terms:**Divide the leading term of the dividend (x^3) by the leading term of the divisor (x). This gives us x^2.**Multiply and subtract:**Multiply the quotient (x^2) by the divisor (x + 2), resulting in x^3 + 2x^2. Subtract this product from the dividend.**Bring down the next term:**Bring down the next term from the dividend (3x).**Repeat steps 2-4:**Repeat the process of dividing, multiplying, and subtracting until you reach a remainder that has a lower degree than the divisor.

### Applying Long Division

Let's perform the long division for (x^3 - 4x^2 + 3x + 2) / (x + 2):

```
x^2 - 6x + 15
x + 2 | x^3 - 4x^2 + 3x + 2
x^3 + 2x^2
-------------
-6x^2 + 3x
-6x^2 - 12x
-------------
15x + 2
15x + 30
-------------
-28
```

### Interpretation of the Result

The result of the long division is:

**Quotient:**x^2 - 6x + 15**Remainder:**-28

This means that:

**(x^3 - 4x^2 + 3x + 2) / (x + 2) = x^2 - 6x + 15 - 28/(x + 2)**

### Conclusion

Long division of polynomials is a powerful tool for dividing polynomials and expressing the result in a more manageable form. This method can be applied to various polynomial expressions and helps us understand the relationship between the dividend, divisor, quotient, and remainder.