Answer
Quotient $=2x^2$
Remainder $=-x^2+x+1$
Work Step by Step
The given expression is
$(4x^5-3x^2+x+1)\div(2x^3-1)$
Rewrite the expression as
$(4x^5+0x^4+0x^3-3x^2+x+1)\div(2x^3+0x^2+0x-1)$
Perform ong division:
$\begin{matrix}
& 2x^2 & & & && & \leftarrow &\text{Quotient}\\
&-- &-- &--&--& \\
2x^3+0x^2+0x-1) &4x^5&+0x^4&+0x^3&-3x^2&+x&+1 & \\
& 4x^5 &+0x^4 &+0x^3 & -2x^2& && \leftarrow &2x^2(2x^3+0x^2+0x-1) \\
& -- & -- & --& --&&& \leftarrow &\text{subtract} \\
& 0 & 0& 0 &-x^2 &+x&+1 & \leftarrow & \text{Remainder}
\end{matrix}$
Checking:
$\text{(Quotient)(divisor)+ Remainder}$
$=(2x^2)(2x^3-1)-x^2+x+1$
$=4x^5-2x^2-x^2+x+1$
$=4x^5-3x^2+x+1$
$=\text{Dividend}$
Hence, the quotient is $2x^2$ and the remainder is $-x^2+x+1$.
