Suppose we have some polynomial $P(x)=a_n x^n+\cdots+a_0$ with coefficients $0\le a_i\le A$ and we want to divide it by $x+b$ where $0<b$. We get a quotient polynomial $Q(x)$ and remainder $R$ so that $P(x)=Q(x)(x+b)+R$. What are some good bounds on the coefficients of $Q$?
If we start looking at the symbolic quotient $a_n x^n+\cdots+a_0-R=(b_{n-1}x^{n-1}+\cdots b_0)(x+b)$ then we can work out the bounds working inwards: $$a_n x^n+\cdots+a_0-R=b_{n-1}x^n + \left((b_{n-1}b+b_{n-2})x^{n-1}+(b_{n-2}b+b_{n-3})x^{n-2}+\cdots+(b_1 b+b_0)x\right) + b_0 b$$ which gives us $0\le b_{n-1}\le A$, $0\le b_k b + b_{k-1}\le A$, and $0\le b_0 b + R\le A$. This gives us "tight" bounds on the first ($b_{n-1}$) and last ($b_0$) coefficients of the quotient, and then we can use the $b_k$ inequality and induction to get bounds for the other coefficients.
Working from left to right, we have $-b_k b\le b_{k-1}\le A - b_k b$. We want to recover bounds for $b_{k-1}$ in terms of the bounds for $b_k$, with the base case being $b_{n-1}$ which has bounds $[0,A]$. Let $[\min(b_k),\max(b_k)]$ be the bounds for $b_k$. Then we see $\min(b_{k-1})=-\max(b_k)b$ and $\max(b_{k-1})=A-\min(b_k)b$. We can solve this using a linear recurrence as $$\begin{bmatrix}\min(b_{k-1})\\\max(b_{k-1})\\A\end{bmatrix} =\begin{bmatrix}0&-b&0\\ -b&0&1\\ 0&0&1\end{bmatrix} \begin{bmatrix}\min(b_k)\\\max(b_k)\\A\end{bmatrix}$$
$$\vec{b}_{k-1} =\begin{bmatrix}\frac{b}{b^2-1}&1&-1\\ \frac{1}{1-b^2}&1&1\\ 1&0&0\end{bmatrix} \begin{bmatrix}1&0&0\\ 0&-b&0\\ 0&0&b\end{bmatrix} \begin{bmatrix}0&0&1\\ \frac{1}{2}&\frac{1}{2}&-\frac{1}{2b+2}\\ -\frac{1}{2}&\frac{1}{2}&\frac{1}{2b-2}\end{bmatrix} \vec{b}_k$$
$$\vec{b}_{k-j} =\begin{bmatrix}\frac{b}{b^2-1}&1&-1\\ \frac{1}{1-b^2}&1&1\\ 1&0&0\end{bmatrix} \begin{bmatrix}1&0&0\\ 0&(-b)^j&0\\ 0&0&b^j\end{bmatrix} \begin{bmatrix}0&0&1\\ \frac{1}{2}&\frac{1}{2}&-\frac{1}{2b+2}\\ -\frac{1}{2}&\frac{1}{2}&\frac{1}{2b-2}\end{bmatrix} \vec{b}_k$$ yielding $$\min(b_{n-1-j})=\frac{1}{2}\left((-b)^j-b^j+\frac{2b-(b-1)(-b)^j-(b+1)b^j}{b^2-1}\right)A$$ $$\max(b_{n-1-j})=\frac{1}{2}\left((-b)^j+b^j+\frac{-2-(b-1)(-b)^j+(b+1)b^j}{b^2-1}\right)A.$$ We can simplify these bounds by using separate expressions for odd and even $j$. We can also do something similar working from right to left to get a different set of bounds and choose the best bounds for each coefficient. Unfortunately, the diagonalization shown is not valid for $b=1$ so that case also needs to be handled separately.
Are there any other (better) known bounds on the coefficients of such a polynomial quotient, or more general polynomial quotients?