**Polynomial Long Division: 6x^5+5x^4+x^3-3x^2+x divided by 3x+1**

In this article, we will perform polynomial long division to divide the polynomial 6x^5+5x^4+x^3-3x^2+x by 3x+1.

**Step 1: Write the dividend and divisor**

The dividend is the polynomial being divided, which is 6x^5+5x^4+x^3-3x^2+x. The divisor is the polynomial by which we are dividing, which is 3x+1.

**Step 2: Divide the leading terms**

To divide the leading terms, we need to find the quotient of the leading terms of the dividend and divisor. In this case, the leading terms are 6x^5 and 3x. The quotient of these terms is 2x^4.

**Step 3: Multiply the divisor by the quotient and subtract**

Multiply the divisor 3x+1 by the quotient 2x^4 and subtract the result from the dividend.

```
2x^4
3x+1 ) 6x^5 + 5x^4 + x^3 - 3x^2 + x
- (6x^5 + 2x^4)
```

**Step 4: Bring down the next term and repeat**

Bring down the next term of the dividend and repeat steps 2 and 3.

```
2x^4 + 1x^3
3x+1 ) 6x^5 + 5x^4 + x^3 - 3x^2 + x
- (6x^5 + 2x^4)
3x^4 + x^3 - 3x^2 + x
- (3x^4 + 1x^3)
-2x^2 + x
```

**Step 5: Repeat the process until the remainder is zero or has a smaller degree than the divisor**

Continue repeating steps 2-4 until the remainder is zero or has a smaller degree than the divisor.

```
2x^4 + 1x^3 - 1x
3x+1 ) 6x^5 + 5x^4 + x^3 - 3x^2 + x
- (6x^5 + 2x^4)
3x^4 + x^3 - 3x^2 + x
- (3x^4 + 1x^3)
-2x^2 + x
- (-2x^2 - 2x)
3x + 1
- (3x + 1)
0
```

**The final result**

The final result of the polynomial long division is:

**Quotient:** 2x^4 + 1x^3 - 1x
**Remainder:** 0

Therefore, we have successfully divided the polynomial 6x^5+5x^4+x^3-3x^2+x by 3x+1 using polynomial long division.