Divide the polynomial `(9x^(2) + 12x + 10)` by (3x + 2) and write the quotient and the remainder.

Answer

To divide the polynomial \(9x^2 + 12x + 10\) by \(3x + 2\), we will use polynomial long division. Here are the steps to find the quotient and the remainder:

### Step 1: Set up the division
We will divide \(9x^2 + 12x + 10\) by \(3x + 2\).

### Step 2: Divide the leading terms
Divide the leading term of the dividend \(9x^2\) by the leading term of the divisor \(3x\):
\[ \frac{9x^2}{3x} = 3x \]
This gives us the first term of the quotient.

### Step 3: Multiply and subtract
Now, multiply \(3x\) by the entire divisor \(3x + 2\):
\[ 3x \cdot (3x + 2) = 9x^2 + 6x \]
Next, subtract this result from the original polynomial:
\[ (9x^2 + 12x + 10) - (9x^2 + 6x) = 12x - 6x + 10 = 6x + 10 \]

### Step 4: Repeat the process
Now, we will divide the new leading term \(6x\) by the leading term of the divisor \(3x\):
\[ \frac{6x}{3x} = 2 \]
This gives us the next term of the quotient.

### Step 5: Multiply and subtract again
Multiply \(2\) by the entire divisor \(3x + 2\):
\[ 2 \cdot (3x + 2) = 6x + 4 \]
Now, subtract this from \(6x + 10\):
\[ (6x + 10) - (6x + 4) = 10 - 4 = 6 \]

### Step 6: Write the final result
At this point, we have completed the division. The quotient is:
\[ 3x + 2 \]
And the remainder is:
\[ 6 \]

### Final Answer:
- **Quotient**: \(3x + 2\)
- **Remainder**: \(6\)

---