Answer
The quotient is $4x^2-11x+23$.
The remainder is $-45$.
Work Step by Step
The given expression is
$(4x^3-3x^2+x+1)\div(x+2)$
Perform synthetic division to obtain:
$\begin{matrix}
& 4x^2 & -11x &+23 & & \leftarrow &\text{Quotient}\\
&-- &-- &--&--& \\
x+2) &4x^3&-3x^2&+x&+1 & \\
& 4x^3 & +8x^2 & & & \leftarrow &4x^2(x+2) \\
& -- & -- & & & \leftarrow &\text{subtract} \\
& 0 & -11x^2 & +x & & \\
& & -11x^2 & -22x & & \leftarrow & -11x(x+2) \\
& & -- & -- & & \leftarrow & \text{subtract} \\
& & 0&23x &+1 & \\
& & & 23x& +46 & \leftarrow & 23(x+2) \\
& & & -- & -- & \leftarrow & \text{subtract} \\
& & & 0 & -45 & \leftarrow & \text{Remainder}
\end{matrix}$
Checking:
(Quotient)(divisor)+ Remainder
$=(4x^2-11x+23)(x+2)-45$
$=4x^3-11x^2+23x+8x^2-22x+46-45$
$=4x^3-3x^2+x+1$
$=$ Dividend
Hence, the Quotient is $4x^2-11x+23$.
and the remainder is $-45$.
