Divide. \( \frac{6 x^{3}+17 x^{2}+20 x+18}{2 x+3} \)

To divide the polynomial \( 6x^3 + 17x^2 + 20x + 18 \) by \( 2x + 3 \), we can use polynomial long division. 1. Divide the leading term of the numerator \( 6x^3 \) by the leading term of the denominator \( 2x \) to get \( 3x^2 \). 2. Multiply \( 3x^2 \) by \( 2x + 3 \) to get \( 6x^3 + 9x^2 \). 3. Subtract \( (6x^3 + 9x^2) \) from \( (6x^3 + 17x^2 + 20x + 18) \). The new polynomial becomes \( 8x^2 + 20x + 18 \). 4. Now, repeat the process: divide \( 8x^2 \) by \( 2x \) to get \( 4x \). 5. Multiply \( 4x \) by \( 2x + 3 \) to get \( 8x^2 + 12x \). 6. Subtract \( (8x^2 + 12x) \) from \( (8x^2 + 20x + 18) \) to get \( 8x + 18 \). 7. Divide \( 8x \) by \( 2x \) to get \( 4 \). 8. Multiply \( 4 \) by \( 2x + 3 \) to get \( 8x + 12 \). 9. Finally, subtract \( (8x + 12) \) from \( (8x + 18) \) to get \( 6 \). Putting it all together, the quotient is \( 3x^2 + 4x + 4 \) and the remainder is \( 6 \). Therefore, \[ \frac{6x^3 + 17x^2 + 20x + 18}{2x + 3} = 3x^2 + 4x + 4 + \frac{6}{2x + 3} \]