Divide p(x) by g(x) in each of the following questions and find the quotient q(x) and remainder r(x) :
`p(x)=x^(3)+3x^(2)+2x+1, " " g(x)=x+2`

Answer

To divide the polynomial \( p(x) = x^3 + 3x^2 + 2x + 1 \) by \( g(x) = x + 2 \), we will use polynomial long division.

### Step-by-Step Solution:

1. **Set up the division**: Write \( p(x) \) under the long division symbol and \( g(x) \) outside.
\[ \begin{array}{r|l} x + 2 & x^3 + 3x^2 + 2x + 1 \\ \end{array} \]

2. **Divide the leading term**: Divide the leading term of \( p(x) \) (which is \( x^3 \)) by the leading term of \( g(x) \) (which is \( x \)):
\[ \frac{x^3}{x} = x^2 \]
This gives us the first term of the quotient \( q(x) \).

3. **Multiply and subtract**: Multiply \( g(x) \) by \( x^2 \) and subtract from \( p(x) \):
\[ (x + 2)(x^2) = x^3 + 2x^2 \]
Now, subtract this from \( p(x) \):
\[ (x^3 + 3x^2 + 2x + 1) - (x^3 + 2x^2) = (3x^2 - 2x^2) + 2x + 1 = x^2 + 2x + 1 \]

4. **Repeat the process**: Now, we divide the new polynomial \( x^2 + 2x + 1 \) by \( g(x) \):
\[ \frac{x^2}{x} = x \]
This gives us the next term of the quotient \( q(x) \).

5. **Multiply and subtract again**: Multiply \( g(x) \) by \( x \) and subtract:
\[ (x + 2)(x) = x^2 + 2x \]
Now, subtract this from \( x^2 + 2x + 1 \):
\[ (x^2 + 2x + 1) - (x^2 + 2x) = 1 \]

6. **Final division**: Now we have a remainder of \( 1 \), and since \( 1 \) is of lower degree than \( g(x) \), we stop here.

### Conclusion:
The quotient \( q(x) \) is \( x^2 + x \) and the remainder \( r(x) \) is \( 1 \).

### Final Answer:
- Quotient \( q(x) = x^2 + x \)
- Remainder \( r(x) = 1 \)