Divide. \[ \left(12 x^{3}-24 x^{2}+6 x+10\right) \div\left(3 x^{2}-3 x\right) \] Your answer should give the quotient and the remainder. Quotient: \( \square \) Remainder: \( \square \)

Divide the polynomials by following steps: - step0: Using Long Division to Divide Polynomials: \(\left(12x^{3}-24x^{2}+6x+10\right)\div \left(3x^{2}-3x\right)\) - step1: Calculate: \(\left(4x-4\right)\text{ }\textrm{remainder}\text{ }\left(-6x+10\right)\) To divide the polynomial \( 12 x^{3}-24 x^{2}+6 x+10 \) by \( 3 x^{2}-3 x \), we perform polynomial long division. 1. **Divide the leading term**: - The leading term of the dividend is \( 12x^3 \) and the leading term of the divisor is \( 3x^2 \). - Dividing these gives \( \frac{12x^3}{3x^2} = 4x \). 2. **Multiply and subtract**: - Multiply \( 4x \) by the entire divisor \( 3x^2 - 3x \): \[ 4x(3x^2 - 3x) = 12x^3 - 12x^2 \] - Subtract this from the original polynomial: \[ (12x^3 - 24x^2 + 6x + 10) - (12x^3 - 12x^2) = -12x^2 + 6x + 10 \] 3. **Repeat the process**: - Now, divide the leading term of the new polynomial \( -12x^2 \) by the leading term of the divisor \( 3x^2 \): \[ \frac{-12x^2}{3x^2} = -4 \] - Multiply \( -4 \) by the divisor: \[ -4(3x^2 - 3x) = -12x^2 + 12x \] - Subtract this from the current polynomial: \[ (-12x^2 + 6x + 10) - (-12x^2 + 12x) = -6x + 10 \] 4. **Final remainder**: - The degree of the remainder \( -6x + 10 \) is less than the degree of the divisor \( 3x^2 - 3x \), so we stop here. Thus, the result of the division is: - **Quotient**: \( 4x - 4 \) - **Remainder**: \( -6x + 10 \) Final answer: - Quotient: \( 4x - 4 \) - Remainder: \( -6x + 10 \)