Divide $3x^{2}-x^{3}-3x+5$ by $x-1-x^{2}$ and verify the division algorithm.

(N/A) Note that the given polynomials are not in standard form. To carry out division,we first write both the dividend and divisor in decreasing order of their degrees.
So,dividend $= -x^{3}+3x^{2}-3x+5$ and divisor $= -x^{2}+x-1$.
Performing the division:
$(-x^{2}+x-1) \overline{) -x^{3}+3x^{2}-3x+5}$
$1$. Divide the first term of the dividend $-x^{3}$ by the first term of the divisor $-x^{2}$ to get $x$. This is the first term of the quotient.
$2$. Multiply the divisor $(-x^{2}+x-1)$ by $x$ to get $-x^{3}+x^{2}-x$. Subtract this from the dividend to get $2x^{2}-2x+5$.
$3$. Divide the first term of the new dividend $2x^{2}$ by the first term of the divisor $-x^{2}$ to get $-2$. This is the second term of the quotient.
$4$. Multiply the divisor $(-x^{2}+x-1)$ by $-2$ to get $2x^{2}-2x+2$. Subtract this from the current dividend to get $3$.
We stop here since the degree of the remainder $(3)$ is $0$,which is less than the degree of the divisor $(-x^{2}+x-1)$,which is $2$.
So,quotient $= x-2$,remainder $= 3$.
Verification:
Divisor $\times$ Quotient $+$ Remainder
$= (-x^{2}+x-1)(x-2)+3$
$= -x^{3}+2x^{2}+x^{2}-2x-x+2+3$
$= -x^{3}+3x^{2}-3x+5$
$= \text{Dividend}$.
Thus,the division algorithm is verified.