Answer
The quotient is $x^4-x^3+6x^2-6x+6$
and the remainder is $-16$.
Work Step by Step
The given expression is:-
$(x^5+5x^3-10)\div (x+1)$
Rewrite as descending powers of $x$.
$(x^5+0x^4+5x^3+0x^2+0x-10)\div (x+1)$
The divisor is $x+1$, so the value of $c=-1$.
and on the right side the coefficients of dividend in descending powers of $x$.
Perform the synthetic division to obtain:
$\begin{matrix}
&-- &-- &--&--& \\
-1) &1&0&5&0&0&-10& \\
& &-1 &1 &-6 &6& -6&\\
& -- & -- & --& -- &--&--& \\
& 1 & -1 & 6 &-6 &6 &-16&\\
\end{matrix}$
The divisor is $x+1$
The dividend is $x^5+5x^3-10$
The Quotient is $x^4-x^3+6x^2-6x+6$
The remainder is $-16$.
Check:-
$\text{(Divisor)(Quotient)+Remainder}$
$=(x+1)(x^4-x^3+6x^2-6x+6)-16$
$=x^5-x^4+6x^3-6x^2+6x+x^4-x^3+6x^2-6x+6-16$
$=x^5+5x^3-10$
Hence, the quotient is $x^4-x^3+6x^2-6x+6$ and the remainder is $-16$.
