What is the quotient of (x^3 + 6x^2 + 11x + 6) divided by (x^2 + 4x + 3)?

1. Set up the polynomial long division: We want to divide the polynomial $x^3 + 6x^2 + 11x + 6$ by the polynomial $x^2 + 4x + 3$. This can be set up like a standard long division problem:

```
____________
x^2+4x+3 | x^3 + 6x^2 + 11x + 6
```

1. Divide the leading terms: Divide the leading term of the dividend ($x^3$) by the leading term of the divisor ($x^2$). This gives $x^3 / x^2 = x$. This is the first term of our quotient.

```
x _________
x^2+4x+3 | x^3 + 6x^2 + 11x + 6
```

1. Multiply the quotient term by the divisor: Multiply the term we just found ($x$) by the entire divisor ($x^2 + 4x + 3$). This gives $x(x^2 + 4x + 3) = x^3 + 4x^2 + 3x$.

```
x _________
x^2+4x+3 | x^3 + 6x^2 + 11x + 6
-(x^3 + 4x^2 + 3x)
```

4. Subtract the result from the dividend: Subtract the product from the dividend. Remember to change the signs of the terms being subtracted:
$(x^3 + 6x^2 + 11x + 6) - (x^3 + 4x^2 + 3x) = x^3 - x^3 + 6x^2 - 4x^2 + 11x - 3x + 6 = 2x^2 + 8x + 6$.

```
x _________
x^2+4x+3 | x^3 + 6x^2 + 11x + 6
-(x^3 + 4x^2 + 3x)
----------------
2x^2 + 8x + 6
```

1. Bring down the next term: In this case, we have already used all terms, so we proceed to the next step with the result of the subtraction.

1. Repeat the process: Now, we treat $2x^2 + 8x + 6$ as our new dividend. Divide the leading term of this new dividend ($2x^2$) by the leading term of the divisor ($x^2$). This gives $2x^2 / x^2 = 2$. This is the next term of our quotient.

```
x + 2
x^2+4x+3 | x^3 + 6x^2 + 11x + 6
-(x^3 + 4x^2 + 3x)
----------------
2x^2 + 8x + 6
```

1. Multiply the new quotient term by the divisor: Multiply the new term ($2$) by the divisor ($x^2 + 4x + 3$). This gives $2(x^2 + 4x + 3) = 2x^2 + 8x + 6$.

```
x + 2
x^2+4x+3 | x^3 + 6x^2 + 11x + 6
-(x^3 + 4x^2 + 3x)
----------------
2x^2 + 8x + 6
-(2x^2 + 8x + 6)
```

8. Subtract again: Subtract the product from the current dividend:
$(2x^2 + 8x + 6) - (2x^2 + 8x + 6) = 0$.

```
x + 2
x^2+4x+3 | x^3 + 6x^2 + 11x + 6
-(x^3 + 4x^2 + 3x)
----------------
2x^2 + 8x + 6
-(2x^2 + 8x + 6)
----------------
0
```

1. State the quotient: Since the remainder is 0, the quotient is the polynomial we found on top, which is $x + 2$.