Write the quotient and remainder when we divide :
`(15 x^(2) + x - 6)` by (3x + 2)

Answer

To solve the problem of dividing the polynomial \( (15x^2 + x - 6) \) by \( (3x + 2) \), we will use polynomial long division. Here are the steps:

### Step 1: Set up the division
We will divide \( 15x^2 + x - 6 \) by \( 3x + 2 \).

### Step 2: Divide the leading terms
Divide the leading term of the dividend \( 15x^2 \) by the leading term of the divisor \( 3x \):
\[ \frac{15x^2}{3x} = 5x \]
This gives us the first term of the quotient.

### Step 3: Multiply and subtract
Now, multiply \( 5x \) by the entire divisor \( (3x + 2) \):
\[ 5x \cdot (3x + 2) = 15x^2 + 10x \]
Next, subtract this result from the original polynomial:
\[ (15x^2 + x - 6) - (15x^2 + 10x) = x - 10x - 6 = -9x - 6 \]

### Step 4: Repeat the process
Now, we will divide the leading term of the new polynomial \( -9x \) by the leading term of the divisor \( 3x \):
\[ \frac{-9x}{3x} = -3 \]
This gives us the next term of the quotient.

### Step 5: Multiply and subtract again
Now, multiply \( -3 \) by the entire divisor \( (3x + 2) \):
\[ -3 \cdot (3x + 2) = -9x - 6 \]
Next, subtract this from the current polynomial:
\[ (-9x - 6) - (-9x - 6) = 0 \]

### Conclusion
Since the remainder is \( 0 \), we conclude that the division is exact.

Thus, the quotient is:
\[ \text{Quotient} = 5x - 3 \]
And the remainder is:
\[ \text{Remainder} = 0 \]

### Final Answer:
- Quotient: \( 5x - 3 \)
- Remainder: \( 0 \)

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