Divide. \[ \left(6 x^{3}+11 x^{2}+16 x+21\right) \div(3 x+4) \] Your answer should give the quotient and the remainder: Quotient:

Let us perform polynomial long division of \[ 6x^3 + 11x^2 + 16x + 21 \] by \[ 3x+4. \] 1. **First step:** - Divide the leading term \(6x^3\) by \(3x\): \[ \frac{6x^3}{3x} = 2x^2. \] - Multiply \(2x^2\) by the divisor: \[ 2x^2(3x+4) = 6x^3 + 8x^2. \] - Subtract this from the dividend: \[ \begin{aligned} (6x^3 + 11x^2) - (6x^3 + 8x^2) &= 3x^2. \end{aligned} \] - Bring down the next term \(16x\): \[ 3x^2 + 16x. \] 2. **Second step:** - Divide \(3x^2\) by \(3x\): \[ \frac{3x^2}{3x} = x. \] - Multiply \(x\) by the divisor: \[ x(3x+4) = 3x^2 + 4x. \] - Subtract this from \(3x^2 + 16x\): \[ (3x^2 + 16x) - (3x^2 + 4x) = 12x. \] - Bring down the next term \(21\): \[ 12x + 21. \] 3. **Third step:** - Divide \(12x\) by \(3x\): \[ \frac{12x}{3x} = 4. \] - Multiply \(4\) by the divisor: \[ 4(3x+4) = 12x + 16. \] - Subtract this from \(12x + 21\): \[ (12x + 21) - (12x + 16) = 5. \] The division yields a quotient and a remainder: - Quotient: \(2x^2 + x + 4\) - Remainder: \(5\) Thus, the result of the division is: \[ \text{Quotient: } 2x^2 + x + 4, \quad \text{Remainder: } 5. \]