The quotient is the outcome of the division one number or expression known as the dividend, by another number or expression known as the divisor
The option that gives the correct quotient is;
[tex]\mathbf{2\cdot x + 1 + \dfrac{2 \cdot x - 5}{x^2 + x + 1}}[/tex]
Where, the quotient is 2·x + 1, and the reminder is 2·x - 5
The reason the above option is correct is as follows:
The given dividend is 2·x³ + 3·x² + 5·x - 4
The divisor is x² + x + 1
By long division of a polynomial we have;
2·x + 1(Quotient)
(2·x³ + 3·x² + 5·x - 4) ÷ (x² + x + 1 )
2·x³ + 2·x² + 2·x
0 + x² + 3·x - 4
[tex]{}[/tex] x² + x + 1
[tex]{}[/tex] 0 + 2·x - 5
Therefore, we have;
[tex]\dfrac{2\cdot x^3 + 3 \cdot x^2 + 5 \cdot x - 4}{x^2 + x + 1} = 2\cdot x + 1 + \dfrac{2 \cdot x - 5}{x^2 + x + 1}[/tex]
Therefore, the correct option is;
[tex]\mathbf{2\cdot x + 1 + \dfrac{2 \cdot x - 5}{x^2 + x + 1}}[/tex]
Learn more about long division of polynomial here:
https://brainly.com/question/4685886
