Question

what is the quotient of $2x^3 + 3x^2 + 5x - 4$ divided by $x^2 + x + 1$? $2x + 1 + \frac{2x - 1}{x^2 + x + 1}$ $2x + 1 + \frac{4x - 4}{x^2 + x + 1}$ $2x + 5 + \frac{12x + 1}{x^2 + x + 1}$ $2x + 1 + \frac{2x - 5}{x^2 + x + 1}$

Explanation:

Step1: Divide leading terms

$\frac{2x^3}{x^2}=2x$

Step2: Multiply divisor by $2x$

$2x(x^2+x+1)=2x^3+2x^2+2x$

Step3: Subtract from dividend

$(2x^3+3x^2+5x-4)-(2x^3+2x^2+2x)=x^2+3x-4$

Step4: Divide new leading terms

$\frac{x^2}{x^2}=1$

Step5: Multiply divisor by $1$

$1(x^2+x+1)=x^2+x+1$

Step6: Subtract to get remainder

$(x^2+3x-4)-(x^2+x+1)=2x-5$

Step7: Write final expression

Quotient + $\frac{\text{Remainder}}{\text{Divisor}}$

Answer:

$2x + 1 + \frac{2x - 5}{x^2 + x + 1}$