what is the remainder when (3x³ - 2x² + 4x - 3) is divided by (x² + 3x + 3)?\n30\n3x - 11\n28x - 36\n28x + 30

Explanation:

Step1: Set up polynomial long - division

We divide $3x^{3}-2x^{2}+4x - 3$ by $x^{2}+3x + 3$. Let $3x^{3}-2x^{2}+4x - 3=(ax + b)(x^{2}+3x + 3)+cx + d$. First, find the leading - term of the quotient. Since the leading term of the dividend is $3x^{3}$ and the leading term of the divisor is $x^{2}$, the leading term of the quotient is $3x$.

Step2: Multiply the divisor by the first - term of the quotient

$(3x)(x^{2}+3x + 3)=3x^{3}+9x^{2}+9x$.

Step3: Subtract from the dividend

$(3x^{3}-2x^{2}+4x - 3)-(3x^{3}+9x^{2}+9x)=3x^{3}-2x^{2}+4x - 3 - 3x^{3}-9x^{2}-9x=-11x^{2}-5x - 3$.

Step4: Find the next term of the quotient

The leading term of $-11x^{2}-5x - 3$ is $-11x^{2}$, and dividing by the leading term of the divisor $x^{2}$ gives $-11$.

Step5: Multiply the divisor by the new term of the quotient

$(-11)(x^{2}+3x + 3)=-11x^{2}-33x - 33$.

Step6: Subtract from the previous result

$(-11x^{2}-5x - 3)-(-11x^{2}-33x - 33)=-11x^{2}-5x - 3 + 11x^{2}+33x + 33 = 28x+30$.

Answer:

D. $28x + 30$