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: Set up long - division

We perform polynomial long - division of $2x^{3}+3x^{2}+5x - 4$ by $x^{2}+x + 1$. First, divide the leading term of the dividend $2x^{3}$ by the leading term of the divisor $x^{2}$. The result is $2x$.

Step2: Multiply and subtract

Multiply $x^{2}+x + 1$ by $2x$ to get $2x^{3}+2x^{2}+2x$. Subtract this from the dividend: $(2x^{3}+3x^{2}+5x - 4)-(2x^{3}+2x^{2}+2x)=x^{2}+3x - 4$.

Step3: Divide again

Divide the leading term of the new dividend $x^{2}$ by the leading term of the divisor $x^{2}$, the result is 1.

Step4: Multiply and subtract again

Multiply $x^{2}+x + 1$ by 1 to get $x^{2}+x + 1$. Subtract this from the new dividend: $(x^{2}+3x - 4)-(x^{2}+x + 1)=2x-5$.
Since the degree of $2x - 5$ is less than the degree of $x^{2}+x + 1$, we stop. The quotient is $2x + 1$ and the remainder is $2x-5$. So the result of the division is $2x + 1+\frac{2x - 5}{x^{2}+x + 1}$.

Answer:

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