Answer
Quotient $=5x^2-11$
Remainder $=x+20$
Work Step by Step
The given expression is
$(5x^4-x^2+x-2)\div(x^2+2)$
Rewrite the expression as
$(5x^4+0x^3-x^2+x-2)\div(x^2+0x+2)$
Perform long division:
$\begin{matrix}
& 5x^2 & -11 & & & & \leftarrow &\text{Quotient}\\
&-- &-- &--&--& \\
x^2+0x+2) &5x^4&+0x^3&-x^2&+x&-2 & \\
& 5x^4 &+0x^3 &+10x^2 & && \leftarrow &5x^2(x^2+0x+2) \\
& -- & -- & --& && \leftarrow &\text{subtract} \\
& 0 & 0& -11x^2 &+x &-2 \\
& & & -11x^2 &0x &-22& \leftarrow & -11(x^2+0x+2) \\
& & & -- & -- & -- & \leftarrow & \text{subtract} \\
& & & 0&x &+20& \leftarrow & \text{Remainder}
\end{matrix}$
Checking:
(Quotient)(divisor)+ Remainder
$=(5x^2-11)(x^2+2)+x+20$
$=5x^4-11x^2+10x^2-22+x+20$
$=5x^4-x^2+x-2$
$=$ Dividend
Hence, the quotient is $5x^2-11$ and the remainder is $x+20$.
