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^(4)+6x^(3)-4x^(2)+2x+1, " " g(x)=x^(2)+3x-1`

Answer

To divide the polynomial \( p(x) = x^4 + 6x^3 - 4x^2 + 2x + 1 \) by \( g(x) = x^2 + 3x - 1 \), we will use polynomial long division.

### Step-by-Step Solution:

1. **Set up the division**: Write \( p(x) \) as the dividend and \( g(x) \) as the divisor.
\[ \text{Dividend: } p(x) = x^4 + 6x^3 - 4x^2 + 2x + 1 \]
\[ \text{Divisor: } g(x) = x^2 + 3x - 1 \]

2. **Divide the leading term**: Divide the leading term of the dividend \( x^4 \) by the leading term of the divisor \( x^2 \):
\[ \frac{x^4}{x^2} = x^2 \]
This is the first term of the quotient \( q(x) \).

3. **Multiply and subtract**: Multiply \( g(x) \) by \( x^2 \) and subtract from \( p(x) \):
\[ x^2 \cdot g(x) = x^2(x^2 + 3x - 1) = x^4 + 3x^3 - x^2 \]
Now subtract:
\[ (x^4 + 6x^3 - 4x^2 + 2x + 1) - (x^4 + 3x^3 - x^2) = (6x^3 - 3x^3) + (-4x^2 + x^2) + 2x + 1 \]
Simplifying this gives:
\[ 3x^3 - 3x^2 + 2x + 1 \]

4. **Repeat the process**: Now, divide the leading term \( 3x^3 \) by \( x^2 \):
\[ \frac{3x^3}{x^2} = 3x \]
Add this to the quotient \( q(x) \).

5. **Multiply and subtract again**: Multiply \( g(x) \) by \( 3x \) and subtract:
\[ 3x \cdot g(x) = 3x(x^2 + 3x - 1) = 3x^3 + 9x^2 - 3x \]
Now subtract:
\[ (3x^3 - 3x^2 + 2x + 1) - (3x^3 + 9x^2 - 3x) = (-3x^2 - 9x^2) + (2x + 3x) + 1 \]
Simplifying gives:
\[ -12x^2 + 5x + 1 \]

6. **Continue the process**: Divide \( -12x^2 \) by \( x^2 \):
\[ \frac{-12x^2}{x^2} = -12 \]
Add this to the quotient \( q(x) \).

7. **Final multiplication and subtraction**: Multiply \( g(x) \) by \( -12 \):
\[ -12 \cdot g(x) = -12(x^2 + 3x - 1) = -12x^2 - 36x + 12 \]
Now subtract:
\[ (-12x^2 + 5x + 1) - (-12x^2 - 36x + 12) = (5x + 36x) + (1 - 12) \]
Simplifying gives:
\[ 41x - 11 \]

8. **Conclusion**: The quotient \( q(x) \) and remainder \( r(x) \) are:
\[ q(x) = x^2 + 3x - 12 \]
\[ r(x) = 41x - 11 \]

### Final Answer:
- Quotient \( q(x) = x^2 + 3x - 12 \)
- Remainder \( r(x) = 41x - 11 \)