Answer:
( x + 2 ) + ( -3x - 3)/(x + 2)
Step-by-step explanation:
(x^3+2x^2 -x +1) / (x^2 + 2)
= ( x + 2 ) + ( -3x - 3)/(x + 2)
I solved this problem by long division, so I do know how to explain it here but if you have any questions, you can ask me in the comments.
Answer:
[tex]x+2-\dfrac{3x+3}{x^2+2}[/tex]
Step-by-step explanation:
As the divisor is quadratic, use long division rather than synthetic division:
[tex]\large \begin{array}{r}x+2\phantom{)}\\x^2+2{\overline{\smash{\big)}\,x^3+2x^2-x+1\phantom{)}}}\\{-~\phantom{(}\underline{(x^3\phantom{))))))}+2x)\phantom{-b)}}\\2x^2-3x+1\phantom{)}\\-~\phantom{()}\underline{(2x^2\phantom{))))))}+4)\phantom{}}\\-3x-3\phantom{)}\\\end{array}[/tex]
Therefore:
[tex]\textsf{Dividend}: \quad x^3+2x^2-x+1[/tex]
[tex]\textsf{Divisor}: \quad x^2+2[/tex]
[tex]\textsf{Quotient}: \quad x+2[/tex]
[tex]\textsf{Remainder}: \quad -3x-x=-(3x+x)[/tex]
When dividing a polynomial, the result is the quotient plus the remainder over the divisor.
[tex]\implies \dfrac{x^3+2x^2-x+1}{x^2+2}=x+2-\dfrac{3x+3}{x^2+2}[/tex]
