What is the quotient of (x^3 + 3x^2 + 5x + 3) divided by (x + 1)?

1. Set up polynomial long division: We want to divide the polynomial $x^3 + 3x^2 + 5x + 3$ by $x + 1$. We set this up like a standard long division problem.

```
____________
x + 1 | x^3 + 3x^2 + 5x + 3
```

1. Divide the leading terms: Divide the leading term of the dividend ($x^3$) by the leading term of the divisor ($x$). This gives $x^3 / x = x^2$. Write this $x^2$ above the $x^2$ term in the dividend.

```
x^2 ________
x + 1 | x^3 + 3x^2 + 5x + 3
```

1. Multiply and subtract: Multiply the quotient term ($x^2$) by the entire divisor ($x + 1$). This gives $x^2(x + 1) = x^3 + x^2$. Write this result below the dividend and subtract it.

```
x^2 ________
x + 1 | x^3 + 3x^2 + 5x + 3
-(x^3 + x^2)
__________
2x^2
```

1. Bring down the next term: Bring down the next term of the dividend ($+5x$) to form the new polynomial to work with.

```
x^2 ________
x + 1 | x^3 + 3x^2 + 5x + 3
-(x^3 + x^2)
__________
2x^2 + 5x
```

1. Repeat the process: Divide the leading term of the new polynomial ($2x^2$) by the leading term of the divisor ($x$). This gives $2x^2 / x = 2x$. Write this $+2x$ in the quotient.

```
x^2 + 2x ____
x + 1 | x^3 + 3x^2 + 5x + 3
-(x^3 + x^2)
__________
2x^2 + 5x
```

1. Multiply and subtract again: Multiply the new quotient term ($2x$) by the divisor ($x + 1$). This gives $2x(x + 1) = 2x^2 + 2x$. Write this below and subtract.

```
x^2 + 2x ____
x + 1 | x^3 + 3x^2 + 5x + 3
-(x^3 + x^2)
__________
2x^2 + 5x
-(2x^2 + 2x)
__________
3x
```

1. Bring down the last term: Bring down the last term of the dividend ($+3$).

```
x^2 + 2x ____
x + 1 | x^3 + 3x^2 + 5x + 3
-(x^3 + x^2)
__________
2x^2 + 5x
-(2x^2 + 2x)
__________
3x + 3
```

1. Final division step: Divide the leading term of the new polynomial ($3x$) by the leading term of the divisor ($x$). This gives $3x / x = 3$. Write this $+3$ in the quotient.

```
x^2 + 2x + 3
x + 1 | x^3 + 3x^2 + 5x + 3
-(x^3 + x^2)
__________
2x^2 + 5x
-(2x^2 + 2x)
__________
3x + 3
```

1. Multiply and subtract one last time: Multiply the new quotient term ($3$) by the divisor ($x + 1$). This gives $3(x + 1) = 3x + 3$. Write this below and subtract.

```
x^2 + 2x + 3
x + 1 | x^3 + 3x^2 + 5x + 3
-(x^3 + x^2)
__________
2x^2 + 5x
-(2x^2 + 2x)
__________
3x + 3
-(3x + 3)
________
0
```

1. State the quotient: Since the remainder is 0, the quotient is the polynomial we found above.