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