Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial :`x^2+3x+1,3x^4+5x^3-7x^2+2x+2`

Explanation

To determine whether the first polynomial \( x^2 + 3x + 1 \) is a factor of the second polynomial \( 3x^4 + 5x^3 - 7x^2 + 2x + 2 \), we will perform polynomial long division.

### Step-by-Step Solution:

1. **Set up the division**: We will divide \( 3x^4 + 5x^3 - 7x^2 + 2x + 2 \) by \( x^2 + 3x + 1 \).

2. **Divide the leading terms**:
- Divide the leading term of the dividend \( 3x^4 \) by the leading term of the divisor \( x^2 \):
...