On dividing `3x^(3)+x^(2)+2x+6` by a polynomial g(x), the quotient and remainder are (3x-5) and (3x+21) respectively. Find g(x).

Explanation

To find the polynomial \( g(x) \) that divides \( 3x^3 + x^2 + 2x + 6 \) given the quotient \( 3x - 5 \) and the remainder \( 3x + 21 \), we will use the polynomial division algorithm. The algorithm states that:

\[ \text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder} \]

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Dividend \( p(x) = 3x^3 + x^2 + 2x + 6 \)
- Quotient \( q(x) = 3x - 5 \)
- Remainder \( r(x) = 3x + 21 \)

2. **Set Up the Division Algorithm:**
According to the division algorithm, we can express the relationship as:
\[ p(x) = g(x) \cdot q(x) + r(x) \]
Plugging in the known values:
\[ 3x^3 + x^2 + 2x + 6 = g(x) \cdot (3x - 5) + (3x + 21) \]

3. **Rearranging the Equation:**
Rearranging gives us:
\[ g(x) \cdot (3x - 5) = (3x^3 + x^2 + 2x + 6) - (3x + 21) \]
Simplifying the right-hand side:
\[ 3x^3 + x^2 + 2x + 6 - 3x - 21 = 3x^3 + x^2 - x - 15 \]
Therefore, we have:
\[ g(x) \cdot (3x - 5) = 3x^3 + x^2 - x - 15 \]

4. **Dividing the Polynomials:**
Now, we need to divide \( 3x^3 + x^2 - x - 15 \) by \( 3x - 5 \) to find \( g(x) \).

- **First Term:** To eliminate \( 3x^3 \), we multiply \( 3x - 5 \) by \( x^2 \):
\[ x^2(3x - 5) = 3x^3 - 5x^2 \]
Subtracting gives:
\[ (3x^3 + x^2) - (3x^3 - 5x^2) = 6x^2 \]
- **Second Term:** Now we have \( 6x^2 - x - 15 \). To eliminate \( 6x^2 \), we multiply \( 3x - 5 \) by \( 2x \):
\[ 2x(3x - 5) = 6x^2 - 10x \]
Subtracting gives:
\[ (6x^2 - x) - (6x^2 - 10x) = 9x - 15 \]
- **Third Term:** Now we have \( 9x - 15 \). To eliminate \( 9x \), we multiply \( 3x - 5 \) by \( 3 \):
\[ 3(3x - 5) = 9x - 15 \]
Subtracting gives:
\[ (9x - 15) - (9x - 15) = 0 \]

5. **Conclusion:**
After performing the division, we find:
\[ g(x) = x^2 + 2x + 3 \]

### Final Answer:
The polynomial \( g(x) \) is:
\[ \boxed{x^2 + 2x + 3} \]