So, in an an exercise regarding polynomial division using the long division method, it asked to prove that $M(x)$ is divisible by $N(x)$
In 3 exercises it gave different values to $M(x)$ and $N(x)$
$M(x)=x^4 - x^3 - 2x^2$
$N(x)=x+1$
$M(x)=x^5+x^4-x^3+2$
$N(x)=x^3-2x+2$
$M(x)=x^5+2x^3+4x^2+x$
$N(x)=x^3-x^2+3x+1$
So, to prove this, after doing long division, the remainder ($R(x)$) should be $0$.
I've tried it over and over again and I dont get a remainder of $0$ in either of the exercises.
So, either the textbook put 3 exercises that are wrong (unlikely), or I made some mistake that I just can't figure out alone.
Here's what I did for exercise 3, for example:
$$\require{enclose} \begin{array}{rll} x^2-x-2 && \hbox{(Explanations)} \\[-3pt] x^3-x^2+3x+1 \enclose{longdiv}{x^5+2x^3+4x^2+x}\kern-.2ex \\[-3pt] \underline{-x^5+x^4-3x^3+x^2-x-2\phantom{00}} && \hbox{($x^2 \times x^3 = x^5$)} \\[-3pt] -x^4-x^3-3x^2+x\phantom{0} && \hbox{($etc$)} \\[-3pt] \underline{\phantom{0}-x^4+x^3-3x^2-x\phantom{0}} \\[-3pt] \phantom{0}-2x^2 \\[-3pt] \underline{\phantom{0}-2x^3+2x^2-6x-2} \\[-3pt] \phantom{00}? \end{array}$$
The remainder here cannot be $0$
I didn't do the other two (or the rest of the explanation of auxiliary calculations) because it's a lot of effort to write long division in $LaTeX$. I hope this information is enough.
Could someone tell me what I did wrong? And maybe post a resolution (to the other 2 problems as well), I don't bother if it is a photo, as long as it is readable (because writing this in LaTeX takes a lot of effort).