Divide $15+2m^4-31m+9m^2+4m^3+m^5$ by $3-2m-m^2$
I use the method of coefficients. I first arrange the dividend and divisor coefficients in descending powers then proceed to the operation:
$-1, -2, 3$ $ \begin{vmatrix} 1 & 2 & 4 &9&-31 &15 \\ 1 & 2 & -3 \\ & & 7 &9&-31& \\ & & 7 &14&-21 \\ & & &-5&-10 &15 \\ & & &-5&-10 &15 \\ \end{vmatrix} $ $-1,-7,5$
The quotient coefficients are $-1, -7, 5$. Since the entire operation is in descending order of powers, I naturally assumed the quotient is $-m^2-7m+5$. But it isn't. The correct quotient is $-m^3-7m+5$. Here is my question. How can I tell what the correct power is? Obviously I can look back at the problem statement and see $-m^2$ and $m^5$ and determine the quotients' first coefficient has to be to the third power, but then since there is only three terms in this quotient how do I determine the power of the second and third terms? Thank you.