Divide the polynomial `16x^(2) + 24x + 15` by (4x + 3) and write the quotient and the remainder.

Answer

To divide the polynomial \( 16x^2 + 24x + 15 \) by \( 4x + 3 \), we will use polynomial long division. Here are the steps:

### Step 1: Set up the division
We write \( 16x^2 + 24x + 15 \) as the dividend and \( 4x + 3 \) as the divisor.

### Step 2: Divide the leading terms
Divide the leading term of the dividend \( 16x^2 \) by the leading term of the divisor \( 4x \):
\[ \frac{16x^2}{4x} = 4x \]
This gives us the first term of the quotient.

### Step 3: Multiply and subtract
Now, multiply \( 4x \) by the entire divisor \( 4x + 3 \):
\[ 4x \cdot (4x + 3) = 16x^2 + 12x \]
Next, subtract this result from the original polynomial:
\[ (16x^2 + 24x + 15) - (16x^2 + 12x) = (24x - 12x) + 15 = 12x + 15 \]

### Step 4: Repeat the process
Now, we take the new polynomial \( 12x + 15 \) and divide its leading term \( 12x \) by the leading term of the divisor \( 4x \):
\[ \frac{12x}{4x} = 3 \]
This gives us the next term of the quotient.

### Step 5: Multiply and subtract again
Multiply \( 3 \) by the entire divisor \( 4x + 3 \):
\[ 3 \cdot (4x + 3) = 12x + 9 \]
Subtract this from the current polynomial:
\[ (12x + 15) - (12x + 9) = 15 - 9 = 6 \]

### Step 6: Write the final result
Now we have completed the division. The quotient is \( 4x + 3 \) and the remainder is \( 6 \).

### Final Answer
- Quotient: \( 4x + 3 \)
- Remainder: \( 6 \)