To find the quotient when [tex]2x^3+x+3[/tex] is divided by [tex]x+1[/tex] you have to do such steps:
1. Multiply [tex]x+1[/tex] by [tex]2x^2[/tex] and subtract the result from the [tex]2x^3+x+3[/tex] in order to eliminate [tex]2x^3[/tex] term:
[tex]2x^3+x+3-2x^2(x+1)=2x^3+x+3-2x^2-2x^2=-2x^2+x+3.[/tex]
2. Multiply [tex]x+1[/tex] by [tex]-2x[/tex] and subtract the result from the [tex]-2x^2+x+3[/tex] in order to eliminate [tex]-2x^2[/tex] term:
[tex]-2x^2+x+3-(-2x)(x+1)=-2x^2+x+3+2x^2+2x=3x+3.[/tex]
3. Multiply [tex]x+1[/tex] by [tex]3[/tex] and subtract the result from the [tex]3x+3[/tex] in order to eliminate [tex]3x[/tex] term:
[tex]3x+3-3(x+1)=3x+3-3x-3=0.[/tex]
4. Now you can write that
[tex]2x^3+x+3=(x+1)(2x^2-2x+3).[/tex]
The quotient is [tex]2x^2-2x+3[/tex] and the remainder is 0.
Answer: correct choice is B
